Please help me with calculus puctures are all attached! But only do these problems

7.2:numbers 3-10 all

7.3: number 3a and b (with graph of the rectangles), 16, 39 (right endpoints only - show the 3 values being summed)

7.4: numbers 2, 3, 4, 6, 9, 13, 21, 57 (find total pollution)

7.2 EXERCISES 1. Integration by substitution is related to what differentiation method? What type of integrand suggests using integration by 9. z V 4z? - 5 dz 10 . rV5r2 + 2 dr substitution? 2. The following integrals may be solved using substitution. 11. 3x e2x dx 12 . re- dr Choose a function u that may be used to solve each problem. Then find du. 13. (1 - 1)e2-F dt 14 . ( x 2 - 1 )er - 3x dx ( a) (3x 2 - 5 ) 4 2x dx (b) VI - x dx el/z x2 15. dz 16. dy (c) ( d) 4x3er dx 2Vy 2x3 + 1 dx -4x 17. dt 18. - dx Use substitution to find each indefinite integral. 12 + 2 x2 + 3 3 . 4 ( 2 x + 3 ) + dx 4. ( - 41 + 1 ) 3 dt x' + 2x 12 + 2 19. dx 20. dt x* + 4x2 + 7 13 + 61 + 3 2 dm 3 du 5. 2x + 1 y- ty (2m + 1)3 6. V3u - 5 21. ( x 2 + x ) 3 dx 22. (2y3 + 3y2 + 1) 213 dy 2x + 2 7. 6x2 dx (x2 + 2x - 4)4 dx 8. (2 x3 + 7 ) 3/2 23. p(p + 1)5 dp 24. 4rv/8 - r dr1. Explain the difference between an indefinite integral and a defi- nite integral. 2. Complete the following statement. (x2 + 3) dx = lim -, where Ax = , and x; is 3. Let f(x) = 2x + 5, x, = 0, X2 = 2, X3 = 4, x4 = 6, and Ax = 2. (a) Find Ef ( x. ) Ax. (b) The sum in part (a) approximates a definite integral using rectangles. The height of each rectangle is given by the value of the function at the left endpoint. Write the definite integral that the sum approximates.404 CHAPTER 7 Integration 14. Consider the region below f(x) = 5 - x, above the x-axis, and between x = 0 and x = 5. Let x; be the midpoint of the ith subinterval. (a) Approximate the area of the region using five rectangles. (b) Find (5 - x) dx by using the formula for the area of a triangle. 15. Find J f(x) dx for each graph of y = f(x). (a) y (b) 16. Find Jf(x) dx for each graph of y = f(x), where f(x) con- sists of line segments and circular arcs. (a) (b) 1 X X7.3 Area and the Definite Integral 407 39. Distance The speed of a particle in a test laboratory was noted every second for 3 seconds. The results are shown in the following table. Use the left endpoints and then the right end- points to estimate the total distance the particle moved in the first three seconds. t (sec) 0 1 2 3 v (ft/sec) 10 6.5 6 5.57.4 EXERCISES Evaluate each definite integral. 33. f(x) = 2 - 2x2; [0, 5] 1. [(- 3 ) ap 2. [Vzax 34. f (x) = 9 - x2; [0, 6] 35. f(x) = x3; [-1, 3] 3. ( 5t - 3 ) dt 4. ( 42 + 3 ) de 36. f (x) = x3 - 2x; [-2, 4] 37. f(x) = e* - 1; [-1, 2 5. ( 5x 2 - 4x + 2) dx 6 . [ ( - x 2 - 3x + 5 ) dx 38. f(x) = 1 - e-*; [-1,2] 7 . 3 V Au + I du 39. f(x) = - - 1; [1, e] ] 8. Var - 2 dr 40. f (x) = 1 - -; [e-', e] D. [2( 132 - 1 ) dt 10 . - ( 3 x 32 + * 12 ) dx Find the area of each shaded region. 11. [(sy Vy + 3V/x) dy 12 . [ ( 4Vr - 3r Vr ) dr 41. 42. Ly=4-12 J. ( 2x - 7 ) 2 dx 14. ( 2 p + 1 ) 2 dp 15 . ( 6n 2 - " - 3 ) an 16 . ( 3 x 3 - 5x 4 ) dx 17 . ( 20- Oly + 3 ) as 18. [ (2 + 320.31) dt -1 0 0 2 f( x) =x2-2x 19. [ (et - 1 -2 - ( u + 1) 2 ) du 20. ( p3 - etp ) dp 21 . > ( 2y 2 - 3 ) 5 dy . m' (4m' + 2) 3 am 23. (64 Vz - 2 dz. 24. ($3 - 1/3 2/3 dy 43. 44. 25. 2 In & dx 26. Vin x dx x x 4+ 27. * 13 V x 43 + 9 dx 28. 3 1 x ( 1 + Inx ) dx y = In x 29. e 21 Jo (3 + ez)2 di 30. -2 - Jo V1+ ezz dz 2 In Exercises 31-40, use the definite integral to find the area y= el - e between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. -2+ 31. f(x) = 2x - 14; [6, 10] 32. f(x) = 4x - 32; [5, 10]Life Sciences 51'. Pollution Pollution from a factor},r is entering a lake. The rate of concentration of the pollutant at time I is given by P'ii) = 140:5\". where r is the number of years since the factory started intro dueing pollutants into the lake. Eeologists estimate that the lake can accept a total level of pollution of 4850 units before all the sh life in the lake ends. Can the factor},r operate for 4 years without killing all the sh in the lake? 53. Spread of an Oil Leak An oil tanker is leaking oil at the rate given [in barrels per hour} by Bln (f + 1:} L' = (f) r+l where r is the time {in hours} after the tanker hits a hidden rook [when r = U}. {a} Find the total number of barrels that the ship will leak on the rst day