Integration. Each correct answer in this section is worth two marks.
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1 PSf Integration Paper 1 Section A 1. Evaluate A. 2 B Each correct answer in this section is worth two marks. 1/2 d. C. 1 2 D. 2 Ke utcome Grade Facilit Disc. Calculator Content Source D 2.2 C NC C13 HSN 086 PSf hsn.uk.net Page 1 Questions marked c SQA
2 PSf 2. The diagram shows the area bounded b the curves = and = 2 2 between = 1 and = 2. PSf = = Represent the shaded area as an integral. A. B. C. D ( ) d ( ) d ( ) d ( ) d Ke utcome Grade Facilit Disc. Calculator Content Source A 2.2 C NC C17 HSN 094 PSf [END F PAPER 1 SECTIN A] hsn.uk.net Page 2 Questions marked c SQA
3 PSf Paper 1 Section B 3. Find ( ) d Find ( ) d Evaluate 2 1 ( ) 2 d. 5 hsn.uk.net Page 3 Questions marked c SQA
4 PSf 6. Find 2 5 d Find the value of 2 1 u u 2 du (a) Find the value of 2 1 ( 4 2) d. 3 (b) Sketch a graph and shade the area represented b the integral in (a). 2 hsn.uk.net Page 4 Questions marked c SQA
5 PSf 9. Evaluate d Find the value of 4 1 d Evaluate 2 1 ( ) d and draw a sketch to illustrate the area represented b this integral. 5 hsn.uk.net Page 5 Questions marked c SQA
6 PSf 12. hsn.uk.net Page 6 Questions marked c SQA
7 PSf 13. A function f is defined b the formula f () = 4 2 ( 3) where R. (a) Write down the coordinates of the points where the curve with equation = f () meets the - and -aes. 2 (b) Find the stationar points of = f () and determine the nature of each. 6 (c) Sketch the curve = f (). 2 (d) Find the area completel enclosed b the curve = f () and the -ais. 4 hsn.uk.net Page 7 Questions marked c SQA
8 PSf 14. hsn.uk.net Page 8 Questions marked c SQA
9 PSf 15. hsn.uk.net Page 9 Questions marked c SQA
10 PSf 16. hsn.uk.net Page 10 Questions marked c SQA
11 PSf 17. hsn.uk.net Page 11 Questions marked c SQA
12 PSf 18. hsn.uk.net Page 12 Questions marked c SQA
13 PSf 19. Functions f and g are defined on the set of real numbers b f () = 1 and g() = 2. (a) Find formulae for (i) f ( g() ) (ii) g ( f () ). 4 (b) The function h is defined b h() = f ( g() ) + g ( f () ). Show that h() = and sketch the graph of h. 3 (c) Find the area enclosed between this graph and the -ais. 4 hsn.uk.net Page 13 Questions marked c SQA
14 PSf 20. The diagram shows a sketch of the graphs of = and = The two curves intersect at A and touch at B, i.e. at B the curves have a common tangent. = A PSf B = (a) (i) Find the -coordinates of the point of the curves where the gradients are equal. 4 (ii) B considering the corresponding -coordinates, or otherwise, distinguish geometricall between the two cases found in part (i). 1 (b) The point A is ( 1, 12) and B is (3, 8). Find the area enclosed between the two curves. 5 Part Marks Level Calc. Content Answer U2 C2 (ai) 4 C NC C4 = 1 3 and = P1 Q4 (aii) 1 C NC 0.1 parallel and coincident (b) 5 C NC C ss: know to diff. and equate 2 pd: differentiate 3 pd: form equation 4 ic: interpret solution 5 ic: interpret diagram 6 ss: know how to find area between curves 7 ic: interpret limits 8 pd: form integral 9 pd: process integration pd: process limits 10 hsn.uk.net Page 14 find derivatives and equate and = 0 4 = 3, = tangents at = 1 3 are parallel, at = 3 coincident 6 (cubic parabola) or (cubic) (parabola) d 8 ( )d or equiv. 9 [ ] 3 or equiv Questions marked c SQA
15 PSf 21. (a) Find the coordinates of the points of intersection of the curves with equations = 2 2 and = (b) Find the area completel enclosed between these two curves. 3 hsn.uk.net Page 15 Questions marked c SQA
16 PSf 22. hsn.uk.net Page 16 Questions marked c SQA
17 PSf 23. hsn.uk.net Page 17 Questions marked c SQA
18 PSf 24. hsn.uk.net Page 18 Questions marked c SQA
19 PSf 25. hsn.uk.net Page 19 Questions marked c SQA
20 PSf 26. The graph of = g() passes through the point (1, 2). If d d = , epress in terms of For all points on the curve = f (), f () = 1 2. If the curve passes through the point (2, 1), find the equation of the curve A curve with equation = f () passes through the point (2, 1) and is such that f () = Epress f () in terms of. 5 hsn.uk.net Page 20 Questions marked c SQA
21 PSf 29. A curve for which d d = passes through the point ( 1, 2). Epress in terms of A curve for which d d = 62 2 passes through the point ( 1, 2). Epress in terms of. 3 hsn.uk.net Page 21 Questions marked c SQA
22 PSf (a) Evaluate π 2 0 cos 2 d. 3 (b) Draw a sketch and eplain our answer. 2 hsn.uk.net Page 22 Questions marked c SQA
23 PSf The graph of = f () passes through the point ( π 9, 1 ). If f () = sin(3) epress in terms of. 4 Part Marks Level Calc. Content Answer U3 C2 4 A/B NC C18, C23 = 1 3 cos(3) P1 Q8 1 ss: know to integrate 2 pd: integrate 3 ic: interpret ( π 9, 1) 4 pd: process 1 = sin(3) d stated or implied b cos(3) 1 = 1 3 cos( 3π 9 ) + c or equiv. 4 c = 7 6 hsn.uk.net Page 23 Questions marked c SQA
24 PSf 35. Differentiate sin 3 with respect to. Hence find sin 2 cos d Evaluate 0 3 ( 2 + 3) 2 d. 4 hsn.uk.net Page 24 Questions marked c SQA
25 PSf 37. hsn.uk.net Page 25 Questions marked c SQA
26 PSf 38. [END F PAPER 1 SECTIN B] hsn.uk.net Page 26 Questions marked c SQA
27 PSf Paper 2 1. Find ( 2 2)( 2 + 2) 2 d, = 0. 4 Part Marks Level Calc. Content Answer U2 C2 4 C CN C14, C12, C c 2001 P2 Q6 1 ss: start to write in standard form 2 pd: complete process 3 pd: integrate 4 pd: integrate a ve inde c hsn.uk.net Page 27 Questions marked c SQA
28 PSf 2. hsn.uk.net Page 28 Questions marked c SQA
29 PSf 3. The parabola shown crosses the -ais at (0, 0) and (4, 0), and has a maimum at (2, 4). PSf 4 The shaded area is bounded b the parabola, the -ais and the lines = 2 and = k. (a) Find the equation of the parabola. 2 k 4 2 (b) Hence show that the shaded area, A, is given b A = 1 3 k3 + 2k Part Marks Level Calc. Content Answer U2 C2 (a) 2 C CN A19 = P2 Q4 (b) 3 C CN C16 proof 1 ic: state standard form 2 pd: process for 2 coeff. 3 ss: know to integrate 4 pd: integrate correctl 5 pd: process limits and complete proof 1 a( 4) 2 a = 1 3 k 2 (function from (a)) k3 + 2k 2 ( ) hsn.uk.net Page 29 Questions marked c SQA
30 PSf 4. A firm asked for a logo to be designed involving the letters A and U. Their initial sketch is shown in the heagon. A mathematical representation of the final logo is shown PSfrag in the coordinate diagram. The curve has equation = ( + 1)( 1)( 3) and the straight line has equation = 5 5. The point (1, 0) is the centre of half-turn smmetr. Calculate the total shaded area (4, 15) ( 2, 15) Part Marks Level Calc. Content Answer U2 C2 7 C CN C units P2 Q8 1 ss: epress in standard form 2 ss: split area and integrate 3 ss: subtract functions 4 pd: process 5 pd: process 6 pd: process 7 ic: use smmetr or otherwise for total area 1 = (...)d or 1 2 (...)d 3 [ (5 5) ( ) ] d or [ ( ) (5 5) ] d 4 ( )d 5 [ ] or depending on chosen integrals hsn.uk.net Page 30 Questions marked c SQA
31 PSf 5. Calculate the shaded area enclosed between the parabolas with equations = = and = PSf. 6 = Part Marks Level Calc. Content Answer U2 C2 6 C CN C P2 Q5 1 ss: find intersections 2 ss: know to find limits 3 ss: know to integrate (upper lower) 4 pd: simplif 5 pd: integrate 6 pd: process limits = = 0, 5 and 5 0 () 3 (( ) ( )) d 4 (5 2 ) d hsn.uk.net Page 31 Questions marked c SQA
32 PSf 6. hsn.uk.net Page 32 Questions marked c SQA
33 PSf 7. hsn.uk.net Page 33 Questions marked c SQA
34 PSf 8. A point moves in a straight line such that its acceleration a is given b a = 2(4 t) 1 2, 0 t 4. If it starts at rest, find an epression for the velocit v where a = dv dt. 4 Part Marks Level Calc. Content Answer U3 C2 4 C NC C18, C22 V = 4 3 (4 t) P2 Q8 1 ss: know to integrate acceleration 2 pd: integrate 3 ic: use initial conditions with const. of int. 4 pd: process solution 1 V = (2(4 t) 1 2 ) dt stated or implied b (4 t) = 2 1 (4 0) c 4 c = A curve for which d ( d = 3 sin(2) passes through the point 5π 12, ) 3. Find in terms of. 4 Part Marks Level Calc. Content Answer U3 C2 4 A/B CN C18, C23 = 3 2 cos(2) P2 Q10 pd: integrate trig function pd: integrate composite function 3 ss: use given point to find c 4 pd: evaluate c sin(2) d stated or implied b cos(2) 3 3 = 3 2 cos( π) + c 4 c = ( 0 4) 10. Find 1 (7 3) 2 d. 2 Part Marks Level Calc. Content Answer U3 C2 2 A/B CN C22, C c 3(7 3) 2000 P2 Q pd: integrate function pd: deal with function of function (7 3) hsn.uk.net Page 34 Questions marked c SQA
35 PSf 11. [END F PAPER 2] hsn.uk.net Page 35 Questions marked c SQA
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