IB Mathematics HL Year 1 Unit 8: Integral Calculus II (Core Topic 7)

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1 IB Mathematics HL Year 1 Unit 8: Integral Calculus II (Core Topic 7) Homework for Unit 8 Lesson 47 Miscelleneous review (integration) 6E: 4 7. Review Set 6A:, 8, 9, 10. Review Set 6B: 3, 5, 7. 7A: 1,, 3, 4. 7B: 1 (Do an five), (Do all), 3, 4. 7C: 1,. 7D: 1,, 3, 5. Review Set 7A:, 3, 4, 6. Lesson 48 Circular function integration (More Review) (A) Prove that for all real numbers n 0, π/ (Hint: note first that 0 π/ 0 sin n d = π/ π/ 0 cos n d = cos n d. π/ 0 0 cos n ( ) d = cos n d. Now tr the substitution u = + π/ and see what happens.) Lesson 49 Volumes of revolution (disks) 8A.1: A.: 1 4. (A) MPOS (Calculus:) #15 (use our calculator!), #83, #88, #94 (calculator). Lesson 50 Volumes of revolution (washers) 8B: 1,, 3, 4. Review Set 8: 1 6.

2 Homework for Unit 8, continued 1. Let R be the regoin enclosed b the curves = 3 and = 4. Lesson 51 Volumes of revolution (shells) (a) Compute the volume of the solid obtained b revolving R about the line =. (b) Compute the volume of the solid obtained b revolving R about the line = 4. Consider the region bounded b the curve = 1( 3 ) and the -ais. Compute the volume of the solid obtained b revolving this region about the -ais. 3. The region below is revolved about the -ais to form a solid of revolution. Find the volume of this solid. = 4 4 = 4. Revolve the same region above about the line = 3 and compute the volume of the resulting solid. 5. Consider the region bounded b the curve = e, = 0, = 0, and =. Compute the volume of the solid obtained b revolving this region about the -ais. 6. Consider the region bounded b the curve = ( 4) (6 ) and the -ais. Compute the volume of the solid obtained b revolving this region about the -ais. 7. Compute the volume obtained b revolving the elliptical region + 1 about the -ais Use integral calculus to compute the volume of a torus (i.e., a doughnut) whose inner radius is r and whose outer radius is R, where R > r.

3 Homework for Unit 8, continued Lesson 5 Trig substitutions 9A: 1 (do them all!),. 9B: 1 3. (A) MPOS (Calculus:) #60. Lesson 53 Integration b parts 9C: 1 5 (it s important to do these all!) (A) MPOS (Calculus:) #4, #30, #47, #55, #67, #71, #74 (b), #90, #98, # Compute these indefinite integrals: ( 5) d + 3, ( + 17) d 6( ), 4 d 4. Lesson 54 Integration b partial fractions. Compute these indefinite integrals: ( 8 + 8) d , d ( 1) ( + 1), ( 3) d ( + 3) 3. ( 3 + 5) d 3. Compute d 4. Compute. ( + 3) 3 5. Compute 4 3 ( + 10) d Compute these indefinite integrals: ( ) d , (A) MPOS (Calculus:) #63. ( + 4) d Lesson 55 Separable differential equations 9D.1: D.: 1 7, 11, 1, 14, 15, 16. (A) MPOS (Calculus:) #5, #1, #16, #78, #81, #91. (B) See the slope field eercises, net page. (These are actuall AP Calculus problems, but ou might find them instructive.) Unit 8 Test

4 Slope Field Eercises. 1. The slope field indicated below most likel depicts the differential equation (A) d d = + (B) d d = + 5 (C) d d = + 5 (D) d d = + 5 (E) d d = + 5. Given the slope field below, what is the most plausible behavior of the solution of the IVP d = f(, ), ( 4) = 4? d (A) lim = (B) lim = 0 (C) lim = + (D) lim = does not eist. (E) lim =

5 3. You are given the differential equation d d = 1, where (0) = 5. For which 5 value(s) of is = 0? 4. Below is sketched the slope field for the differential equation d = f(, ). Sketch d a possible solution of the above differential equation satisfing the initial value ( 5) = Given the slope field indicated below, sketch solutions of the IVPs (i) d = f(, ), ( 1) = 4, 1 d (ii) d = f(, ), ( 1) = 1, 1 d

6 6. Sketch the slope field describing the solutions of the ODE d d =. 7. Sketch the slope field describing the solutions of the logistic differential equation dp = P (5 P ), P 0. dt P t 8. Consider the differential equation d d = 3. (a) Let = f() be the particular solution to the given differential equation for 1 < < 5 such that the line = is tangent to the graph of f. Find the - coordinate of the point of tangenc, and determine whether f has a local maimum, local minimum, or neither at this point. Justif our answer. (b) Let = g() be the particular solution to the given differential equation for < < 8, with the initial condition g(6) = 4. Find = g().

7 9. Consider the differential equation d d = ( 1). (a) On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated (b) While the slope field in part (a) is drawn at onl twelve points, it is defined at ever point in the -plane. Describe all points in the -plane for which the slopes are positive. (c) Find the particular solution = f() to the given differential equation with the initial condition f(0) = 3. Consider the differential equation d d = 3 e. (a) Find a solution = f() to the differential equation satisfing f(0) = 1. (b) Find the domain and range of the function f found in part (a). ( 10. The function f is differentiable for all real numbers. The point 3, 1 4 ) is on the graph of = f(), and the slope at each point (, ) on the graph is given b d d = (6 ). ( (a) Find d and evaluate it at the point 3, 1 ). d 4 (b) Find = f() b solving the differential equation d d = (6 ) with the initial condition f(3) = 1 4.

8 11. Let f be the function satisfing f () = f() for all real numbers, where f(3) = 5. (a) Find f (3). (b) Write an epression for = f() b solving the differential equation d d = with the initial condition f(3) = Consider the differential equation d d =. (a) On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated (b) Let = f() be the particular solution to the differential equation with the initial condition f(1) = 1. Write an equation for the line tangent to the graph of f at (1, 1) and use it to approimate f(1.1). (c) Find the particular solution = f() to the given differential equation with the initial condition f(1) = 1.