8 FURTHER CALCULUS. 8.0 Introduction. 8.1 Implicit functions. Objectives. Activity 1

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1 8 FURTHER CALCULUS Chapter 8 Further Calculus Objectives After studing this chapter ou should be able to differentiate epressions defined implicitl; be able to use approimate methods for integration such as the trapezium rule and Simpson's rule; understand how to calculate volumes of revolution; be able to find arc lengths and areas of surfaces of revolution; be able to derive and use simple reduction formulae in integration. 8. Introduction You should have alread covered the material in Pure Mathematics, Chapters 8,,, 4, 7 and 8 on calculus. This chapter will enable ou to see applications of the ideas of integration ou have alread met and help ou to find derivatives and values of integrals when previousl ou had no method available. 8. Implicit functions When a curve is defined b a relation of the form = f( ) we sa that is an eplicit function of and we can usuall appl one of the standard procedures to find d. However, the epression = 4 is an implicit function of. It also defines a curve, but because we cannot easil make the subject we need to adopt a different strateg if we wish to find the gradient at a particular point. Let us consider a more simple curve first of all. Activit Sketch the circle with equation + = 5. The point P(, ) lies on the circle and O is the origin. Write down the gradient of OP. Deduce the gradient of the tangent to the circle at P.

2 Suppose z =, then b the Chain Rule dz = dz d d = d Activit B differentiating each term with respect to, show that when + = 5, d =. Hence deduce the value of the gradient at the tangent to the circle + = 5 at the point (, ). Check our answer with that from Activit. Eample Find the gradient of the curve with equation = 4 at the point (, 3). Solution B the product rule, d ( d )= + Also d ( 3 )=3 d So differentiating = 4 with respect to gives When = and = 3, 3 d d = 7 d d = 9 d = d = 9

3 Eample A curve has equation = and P is the point (, 3). (a) Show that P lies on the curve. (b) Show that the curve has a stationar point at P. (c) Find the value of d at P and hence determine whether the curve has a maimum or minimum point at P. Solution (a) Substituting =, = 3 into the right-hand side of the equation gives =. Hence P lies on the curve. (b) Differentiating implicitl with respect to gives d + 3 d = Substituting =, = 3 gives d d + 7 = So d = at the point P. (c) You need to differentiate the first equation in the solution to (b) implicitl with respect to. This gives 6 + d + d + d + 6 d d + 3 d = At P, =, = 3 and d =, so 6 + d + 7 d = d = 3 4 This negative value of d at the stationar point P tells us we have a maimum point at P. 3

4 Eercise 8A Find the gradient of the following curves at the points indicated in Questions to = 3 at (, ) = 5 at (,3) = 6 at ( 3,) 4. cos( + ) + = at (,) 5. e3 + = at (,3) 6. + = 3 at ( 4,) 7. Given that e = e + e, show that d + d =. (AEB) 8. Find the value of d and the value of d at the point (, ) on the curve with equation = Approimate integration trapezium rule Activit 3 Evaluate 4. 3 Now consider the curve with equation = 4. The points A and B on the -ais are where = and = 3 respectivel. The lines = and = 3 cut the curve at D and C respectivel. Write down the -coordinates of D and C. Find the area of the trapezium ABCD and compare our answer with the integral ou evaluated. D A = 4 C B Wh is the approimation quite good? Activit 4 Repeat the idea of Activit 3 for the integral 4 and the corresponding trapezium. Do ou get a good approimation for the integral this time b considering a trapezium? Can ou eplain wh? 4

5 In general, when the graph of = f( ) is approimatel linear for a b, the value of the integral b a ( ) f can be approimated b the area of a trapezium. This can be seen easil from the diagram opposite for the case when f( ) for a b. The area of the trapezium ABCD is b a ( ) f a { ( )+f( b) } A D a = f () C B b Hence b a f( ) ( b a) { f( a)+ f( b) } The curve can usuall be approimated to a straight line if the interval considered is ver small. To obtain an approimation over a larger interval, the interval is usuall split into smaller ones. For ease of computation, the interval of integration is usuall divided into strips of equal width. Activit 5 B considering four strips of equal width and considering the approimate area to be that of four trapezia, estimate the value of cos, working with four decimal places and giving our final answer to three significant figures. (Remember to use a radian setting on our calculator!) How close were ou to the eact value cos+ sin ? In general, if ou wish to find the approimate value of f( ), a ou divide the interval a b into n strips of equal width h. Let ( = a),,,..., n ( =b) be the equall spaced -coordinates. Then the values f( ), f( ),..., f( n ) can be written more convenientl as f, f, f,..., f n and these values are called the ordinates. b 5

6 Activit 6 Given that f( ) for a b, the integral f( ) is an a area. B considering the strips on the diagram opposite to be approimatel trapezia, show that the value of the integral above is approimatel Hence show that h ( f + f )+ h ( f + f )+ K+ h ( f n + f n ). b f( ) h f + ( f + f + K+ f n )+ f n a { } This result is known as the trapezium rule. b f f f f n f n h a = n n = b Sometimes an integral is ver difficult or impossible to evaluate eactl and so an approimate method is used. Eample Use the trapezium rule with 5 ordinates to find the approimate value of 3, + 3 ( ) giving our answer to three decimal places. Solution We must remember that 5 ordinates means 4 strips (rather like 5 fence posts and 4 strips of fencing between them). In this case ( = a)= and 4 ( = b)=3. Each strip is of width.5 and so h =.5; =. 5 ; =.; 3 =.5. Therefore f.77; f.478; f.3333; f 3.453; f Using the trapezium rule, integral.5.75 { f + ( f + f + f 3 )+ f 4 } 6

7 Activit 7 If ou have a graphics calculator with the facilit to graph and integrate, find the value given for 3 ( + 3 ) and compare our answer with that in the eample above. Suggest a wa that a more accurate approimation to the integral in the above eample could be found. Write a computer program to find the approimate value of the integral using the trapezium rule where ou input the number of strips and consider how man strips are necessar in order to give the correct value to a certain number of decimal places. 8.3 Simpson's rule Activit 8 Let f ( ) be approimated b the quadratic function a + b + c over the interval h h. Show that h h f( ) h ( 3 ah + 6c) Activit 9 Given that the quadratic function with equation = a + b + c passes through the points ( h, f ), (, f ) and ( h, f ), show that f + 4 f + f = ah + 6c Suppose that the region bounded b the curve = f( ), the lines = h and = h and the -ais is divided into two strips as shown in the diagram. You can use the the results from Activities 8 and 9 to show that, provided f ( ) is approimatel a quadratic, h h f f f 7

8 h f ( ) h 3 f + 4 f + f h ( ) Since the curve = f( ) has been approimated b a quadratic curve the result above is usuall a better approimation than that given b the trapezium rule which approimates the curve = f( ) b a straight line. Activit Suppose that f, f and f are the ordinates at = a, = a + h and = a + h respectivel. Eplain, using the previous result and a translation of the aes, wh f ( ) h ( 3 f + 4 f + f ). a a+h B taking several pairs of strips, the result from Activit can now be etended. If ou divide the interval a b into n equal strips of width h as before, and again let f, f,..., f n, be the corresponding ordinates, the above rule applied to each pair of strips in turn gives b f( ) h a 3 { ( f + 4 f + f )+( f +4f 3 + f 4 )+ ( f f 5 + f 6 ) ( f n + 4 f n + f n )} or, as the formula is more usuall written, n ( ) f [ ( )] 3 h f + f n +4( f + f 3 + K+f n )+ f + f 4 + K+f n or, as some find it easier to remember h ['ends'+ 4 'odds'+ 'evens' ]. 3 This result is know as Simpson's rule. Note: n must be even, giving an odd number of ordinates. It is sometimes interesting to compare the accurac using Simpson's rule with that from the corresponding application of the trapezium rule. Earlier, the trapezium rule with 5 ordinates was used to find the approimate value of 8

9 3, + 3 ( ) and the value.75 was obtained. Eample Use Simpson's rule with 5 ordinates to find the approimate value of 3, + 3 ( ) giving our answer to three decimal places. Solution As in the previous eample, =, =. 5, =., 3 =.5, 4 = 3. and h =.5. Also, f =.77, f =.478, f =.3333, f 3 =.453, f 4 =.89, working to four decimal places. Using Simpson's rule, the approimate value of the integral is [ ] 3.5 f + f ( f + f 3)+f = [ ].743. This answer is in fact correct to three decimal places. Eercise 8B. Use the trapezium rule with 3 ordinates to 5 estimate ( + 7). 5 Evaluate b integration ( + 7). Eplain our findings.. Estimate the value of each of the following definite integrals () to (4) using (a) the trapezium rule; (b) Simpson's rule each with (i) 3 ordinates, (ii) 5 ordinates. () 3 ( ) 6 () 3 3 (3) ln (4) 4 4 ( ) 3. Evaluate each of the integrals in Question eactl and comment on the accurac of the approimations. 9

10 8.4 Volumes of revolution Integration is a ver powerful tool for finding quantities like areas, volumes and arc lengths. This section will illustrate how integration is used for one particular application, namel the determination of volumes. First, though, some approimate methods will be used. The diagram opposite shows a cone with radius unit, height unit (not to scale). What is its volume? To find its volume ou would probabl use the formula V = 3 r h V but how do ou know this formula is correct and wh the one third factor? One wa to approimate the volume of the cone is to consider it to be made of a series of clinders. Two methods are shown opposite. V clearl over-estimates the volume, whereas V clearl under-estimates the volume. However we do know that the true volume is 'trapped' between the volume V and the volume V. V Activit (a) Calculate an approimation of the volume of a cube b 'averaging' the volumes of V and V where V = A+ B+ C+ D and where V = A'+ B' + C'. D.5 (b) What is the ratio average V = ( ) =? V + V Now repeat with clinders of height: (i). (ii). (iii).5 C B A In each case, calculate the ratio V + V ( ) C'.5 As the clinder height approaches zero what does the ratio ( V + V ) B' A'..5.5 approach?

11 Activit Find an approimation of the volume of a sphere b considering it to be made of a series of clinders as shown opposite. Consider the cone again. Diagram shows the line = from = to =. (, ) If ou rotate this area about the -ais through or 36 ou will generate a cone with radius and length (or height) (diagram ). Now consider a thin slice of the cone of radius and length δ. This will be the slice obtained b rotating the shaded strip shown in diagram 3. Diagram The volume of this slice is approimatel that of a clinder with volume δ and the smaller we make δ the better the approimation becomes. The volume of the cone therefore is Σ δ Diagram where the Σ sign means summing up over all such slices. (, ) Another wa of epressing this is to write δv δ where δv is the volume corresponding to the small clindrical disc of height δ. Hence δ Diagram 3 δv δ and letting δ gives dv =. Integrating gives an epression for the volume V =

12 Activit 3 Evaluate the epression for V given above, noting that = in this case. = f () The formula for the volume of revolution of an curve with equation = f b ( ) about the -ais between = a and = b is given b V = a a b Activit 4 The diagram opposite shows part of the line = m between = and = h. The region bounded b the lines = m, = and = h is to be rotated about the -ais to generate a cone. = m h r (a) Epress m in terms of h and r (radius of the cone). (b) B evaluating V =, show that the volume of a cone is given b h V = 3 r h What would happen if the line = m in the above diagram was rotated about the -ais? Activit 5 The diagram opposite is a model of a power station cooling tower. It is necessar to find its volume. The diagram below it shows part of the curve = e. If the region bounded b the curve = e and the lines = a and = b is rotated about the -ais, a shape which is an approimation to the model of the cooling tower will be generated. Show that the volume generated is given b [ eb e a ] a b Hence calculate the volume when a = and b = 5.

13 Activit 6 (a) In order to estimate the amount of liquid that a saucer will hold, a student decides to use the function = 3. (b) She rotates the area bounded b the line = and the line 3 = about the -ais. Calculate the volume generated. Obtain a 'saucer' of our own and suggest possible improvements to the mathematical model used b this student. δ (c) Obtain a 'cup' of our own and, b producing a mathematical model, estimate its volume. Compare our estimated volume with the actual volume. Modif our model to give a better approimation. Eample The area given b and is to be rotated about the -ais to form a 'bowl'. Find the volume of the material in the bowl. (, ) Solution The points of intersection are (, ) and (, ). You can think of the bowl as the solid formed b the outer curve with the inner curve taken awa. Therefore V = = = ( ) = 3 3 = 6 cubic units. 3

14 Eercise 8C. Find the volume of the solid generated when the region bounded b the -ais, the lines = and = and the curve with equation =, is rotated once about the -ais.. Sketch the curve with equation = 4. The region R is bounded b the curve = 4, the -ais and the lines = and =. Find the volume generated when R is rotated completel about the -ais. 3. The area bounded b the curve with equation 6. A curve has equation = cosh and the points P and Q on the curve have -coordinates and ln respectivel. The region bounded b the arc PQ of the curve, the coordinate aes and the line = ln, is rotated through radians about the -ais to form a solid of revolution. Show that the volume of the solid is ln + 5 ( 3 ). (AEB) 7. B using suitable approimations for the function to represent the shape for a milk bottle, use integration to find the volume of revolution. Check this result with the epected value. = tan, the -ais and the line = 3 is rotated about the -ais. Calculate the volume of the solid of revolution so formed. 4. Draw a rough sketch of the circle with equation + = and the curve with equation 9 =. Find the coordinates of the points A and B where the meet. Calculate the area of the region R bounded b the minor arc AB of the circle and the other curve. Find also the volume obtained b rotating R about the -ais. 5. The region bounded b the curve with equation = +, the -ais and the lines = and 8. The finite region bounded b the curves with equations = 3 and = is R. (a) Determine the area of R. (b) Calculate the volume generated when R is rotated through radians about (i) the -ais, (ii) the -ais. = 4, is R. Determine (a) the area of R, (b) the volume of the solid formed when R is rotated through radians about the -ais. (AEB) 8.5 Lengths of arcs of curves Suppose P and Q are two points fairl close together which lie on the curve with equation = f( ), so that P has coordinates (, ) and Q has coordinates ( + δ, + δ). Let the length of the curve between P and Q be δs. This length must be approimatel equal to the length of the line segment PQ. Thus P Q δs (( δ) + ( δ) ) So δs δ δ + δ 4

15 Taking the limit as δ, ds d + Hence the length of arc of curve from the point where = to the point where = is s = + d Eample Find the length of the arc of the curve with equation = from the point where = 3 4 to the point where =. Solution Here d =, so + d = + 4 Arc length s = ( ) 4 = ( )3 34 = 6 ( 7 8) = 3 6 Activit 7 Show that if and are epressed in terms of a parameter t, then the arc length between the points on the curve where t = t and t = t is given b t ( ) s = ẋ + ẏ t dt where ẋ = dt and ẏ = d dt 5

16 Eample A curve is defined b = t sin t; = cost where t is a parameter. Calculate the length of the arc of the curve from the point where t = to the point where t =. Solution = dt = cost; ẏ = d dt = sin t ẋ + ẏ = cost + cos t + sin t But cos t + sin t = giving ẋ + ẏ = cost = 4sin t As sin t for t, ( ẋ + ẏ ) =sin t ( ) and s = ẋ + ẏ = sin t dt = [ 4cos t ] dt = 4cos + 4cos = = 8 Eercise 8D. Use integration to find the length of the arc of the curve with equation = ( ) from the point where = to the point where =. B identifing the curve, verif our answer.. A curve is defined parametricall b = cost cost, = sint sin t, where t is a parameter. Find the length of the arc of the curve from the point where t = to the point 3. Sketch the curve defined b the parametric equations = cos 3 t; = sin 3 t ( t ) using a graphics calculator or software package. Show that the total length of the curve is 6 units, b making use of smmetr properties of the curve. Wh can ou not find the arc length b integrating directl from t = to t =? where t = 4. 6

17 4. Given that = lnsec, where < <, show that d = tan. Hence find the length of the curve with equation = lnsec from the point where = 4 to the point where = Show that the length of arc of the parabola with parametric equations = at, = at (where a>), from the point where t = t to the point where t = t, is given b t ( ) a + t dt. t B means of the substitution t = sinhθ, or otherwise, show that the arc length from the point where t = to the point where t = 3 is a( sinh ). 6. Calculate the length of the curve defined b the equations = tanht, = secht, from the point where t = to the point where t =. 7. Given that a is a positive constant, find the length of the curve defined parametricall b = atsint ( t 3). = at cost; 8. The curve defined parametricall b = ( sint)cost, = ( sint) sint is sketched for t. It is called a cardioid. What happens when ou tr to find the total perimeter b integrating from to? Use smmetr properties to find the perimeter of the cardioid. 8.6 Curved surface areas of revolution Consider the small arc PQ of length δs of a curve distance from the -ais. When the arc is rotated through radians about the -ais it generates a band with surface area δa approimatel equal to δs. P Q Therefore δa δ δs δ Taking the limit as δ and using the formula for ds obtained in the previous section gives da = ds = + d Hence the curved suface area of revolution when the arc of a curve from = to = is rotated through radians about the -ais is + d 7

18 Eample The arc of the parabola with equation = from the origin to the point (4, ) is rotated through radians about the -ais. Calculate the area of the curved surface generated. Solution = d = + d = + = + + = Since the arc is on the branch where is positive, = ( ) and so curved surface area 4 = + d 4 = ( ) + = ( +) 4 [ ] 4 = ( 3 + )3 = 3 ( 7 ) = 5 3 The results from Activit 7 give a corresponding result when a curve is epressed in parametric form with parameter t, as below. Curved surface area is t t [ ẋ + ẏ ] dt 8

19 Activit 8 A curve is defined parametricall b = t sin t, = cost. Show that ẋ + ẏ = 4sin t. The part of the curve from t = to t = is rotated through radians about the -ais to form a curved surface. Show that the surface area can be epressed in the form 8 sin 3 t dt Use the substitution u = cos t, or otherwise, to show that this surface area has value 3 3. Eercise 8E. The arc of the curve described b = cos 3 t, = sin 3 t, from the point where t = to the point where t =, is rotated through radians about the -ais. Calculate the area of the surface generated.. Calculate the area of the surface generated when each of the following arcs of curves are rotated through radians about the -ais. (a) the part of the line = 4 from = to = 5 ; (b) the arc of the curve = cosh from = to = ; 3. The parametric equations of a curve are = t tanht, = secht The arc of the curve between the points with parameters and ln is rotated about the -ais. Calculate in terms of the area of the curved surface formed. (AEB) 4. A curve is defined b = t 3, =3t where t is a parameter. Calculate the area of the surface generated when the portion of the curve from t = to t = is rotated through radians about the -ais. (c) the portion of the curve = e from = to = ln3. 9

20 8.7 Reduction formulae in integration Activit 9 Use integration b parts to show how I = e can be epressed in terms of e. Evaluate e and hence find the value of I. If ou consider the more general form of this integral I n = n e ou can use integration b parts to epress I n in terms of I n. Now putting u = n and dv = e gives Assuming n >, [ ] + n n I n = n e e I n = e +ni n This is an eample of a reduction formula. Suppose ou wish to find the value of the integral in Activit 9. You need I. But I = e +I and I = e +I But Hence I = e = [ e ] = e I = e +I = e and I = e + e ( )= 5e 3

21 Activit Use the reduction formula and value of I to obtain the values of 3 e and 4 e Eample Given that I n = sin n, epress I n in terms of I n for n. Hence evaluate sin 5 and sin 6. Solution Using the formula for integration b parts u dv = uv du let u = sin n, dv = sin, v giving du = n ( )sinn cos, v = cos. So for n [ ] I n = sin n cos = + n = n ( ) sin n cos + n ( ) sin n sin ( ) ( ) sin n ( n ) = ( n )I n ( n )I n Therefore for n ni n = ( n )I n sin n or I n = ( n ) I n. n 3

22 You need I 5 = 4 5 I 3 and I 3 = 3 I [ ] Also I = sin = cos = and I 5 = = 8 5. Similarl, I 6 = I = Activit (a) (b) Differentiate sec n tan with respect to and epress our answer in terms of powers of sec. B integrating the result with respect to, establish the reduction formula ( n )I n ( n )I n =sec n tan + constant where I n = sec n. Use the reduction formula in (a) to evaluate 4sec 4 4 and sec 5. 6 Eercise 8F. Given that I n = cos n, prove that I n = n n I n. Hence find I 4 and I 7.. Given that I n = n cos, establish the reduction formula ( n + 4) I n = nn ( )I n +e. Evaluate I 4 and I 5. ( n + 4) I n = nn 3. If I n = e sin n, n >, show that ( )I n +e Hence, or otherwise, find e sin 3. (AEB) 4. If I n = tan n, obtain a reduction formula for I n. Hence, or otherwise, show that 4 5. Simplif If I n = = 3 8. (AEB) sin nθ sin ( n )θ. sinθ tan 4 I n = I n + sin nθ dθ, prove that sinθ n ( ) sin( n )θ. Hence, or sin5θ otherwise, evaluate dθ. (AEB) sinθ 6. Given that p = ln, write down the values of cosh p and sinh p. Find a reduction formula p relating I n and I n- where I n = cosh n θ dθ. Hence find I 3 and I 4. 3

23 8.8 Miscellaneous Eercises. Sketch the curve given b the parametric equations = t 3 ; = 3t. Find the length of the arc of the curve from t = to t =. (AEB). Given that a is a positive constant, find the length of the arc of the curve with equation = acosh between = and = k, where k is a a constant. Hence show that, as k varies, the arc length is proportional to the area of the region bounded b the arc, the coordinate aes and the line = k. 3. Given that = 3 +, find d, giving our answer in terms of and. (AEB) 4. Find the length of the arc l of the curve with equation = ln cos ( ) from the point at which = to the point at which =. The arc l is 6 rotated completel about the -ais to form a surface, S. Show that the area of S is 6 ln( cos ) cos Use Simpson's rule with three ordinates to estimate this area, giving our answer to decimal places. (AEB) 5. Given that I =, ( 4 ) (a) using Simpson's rule with four equal intervals, working with 4 decimal places, obtain an approimation to I, giving our answer to three significant figures; (b) b putting = sinθ, show that 6 I = 4 sin θ dθ. Chapter 8 Further Calculus 7. A curve has equation = + +. Calculate the coordinates of the turning points of the curve and determine their nature. The finite region bounded b the curve, the -ais from = to = 4 and the line = 4 is R. Epress the area of R as an integral and show that its eact value is 4 + ln5. Use Simpson's rule with four equal intervals to find an estimate of the same integral, giving our answer to three decimal places. Hence find an approimation for ln5, giving our answer to two decimal places. (AEB) 8. Shade on a sketch the finite region R in the first quadrant bounded b the -ais, the curve with equation = ln and the line = 5. B means of integration, calculate the area of R. The region R is rotated completel about the -ais to form a solid of revolution S ( ln ) Use the given table of values and appl the trapezium rule to find an estimate of the volume of S, giving our answer to one decimal place. (AEB) 9. Showing our working in the form of a table, use Simpson's rule with 4 equal intervals to estimate ln( + ), giving our answer to 3 decimal places. Deduce an approimate value for ln ( + ). (AEB). Given that > and that = ln d, find. State the set of values of for which d > and the set of values of for which that has a maimum value of e. d <. Hence show Hence show that the eact value of I is 3 3. (AEB) 6. Find the gradient at the point (, 3) on the curve with equation =. (AEB) Find the area of the finite region R bounded b the curve = ln, the -ais and the line = 5. The region R is rotated completel about the -ais to form a solid of revolution S. Use Simpson's rule, taking ordinates at =,, 3, 4 and 5 to estimate, to significant figures, the volume of S. (AEB) 33

24 . The following approimate measurements were made of two related variables and Use the trapezium rule with five ordinates to 4 estimate the value of giving our answer to decimal place. (AEB) 5. Evaluate + giving our answer in terms of a natural logarithm. Use Simpson's rule with four intervals to find an estimate of the integral. Hence find an approimation for ln3, giving our answer to two decimal places. (AEB) 4 3. Given that I n = tan n, show that for n, I n+ + I n = n +. Calculate the values of I and I. 4 5 (AEB) 4. A curve has equation 9a = 8 3, where a is a positive constant. The tangent at P a, a 3 meets the curve again at Q. Prove that the -coordinate of Q is 4 a and show further that QP is the normal to the curve at Q. Show that + d = +. Find, in terms of a, the length a of the arc QP of the curve. (AEB) 5. Sketch the curve with equation = sec for. The region bounded b the curve, the -ais and the lines = 6 and = 6 d Show that prove that the perimeter of R is 4 + is R. + = ( sec ) and hence 3. (AEB) 6. A curve has equation = cosh and the points P and Q on the curve have -coordinates and ln respectivel. Find the length of the arc PQ of the curve. (AEB) ( ) 7. Given that I n = n n >, then ( n + 3) I n = ni n. A curve has equation = 4 ( )., prove that, if (a) Find the coordinates of the turning point of the curve and sketch its graph. (b) Find the area of the loop of the curve. (AEB) 8. Derive a reduction formula for I n in terms of I n ( ) n. when I n = 3 ln Hence find 3 ln ( ) 3. (AEB) 9. The tangent at a point P on the curve whose parametric equations are = at ( 3 t3 ), = at, cuts the -ais at T. Prove that the distance of the point T from the origin O is one half of the length of the arc OP. (AEB). A curve is given b the parametric equations = 4cost + cost, = sin t + 4sint + t. Find the length of the curve between the points t = and t = 4. (AEB). A curve has parametric equations = 4t ; = t 4 4lnt. Find the length of the arc of the curve from t = to t =. Find also the area of the surface formed when this arc is rotated completel about the -ais. (AEB). Given that I n =, where a is a ( a + ) constant, show that for n n ni n = n ( a + ) ( n )a I n + constant. 3 5 Evaluate and ( ) 4 ( ) 34