Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln, 4, evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. (Give your answer correct to si decimal places.) What does the Riemann sum represent? Illustrate with a diagram. 3. If f() =, 6,find the Riemann sum with n = 5 correct to si decimal places, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram. 4. Use the Midpoint Rule with the given value of to approimate the integral Round the answer to four decimal places. 5. Epress the limit lim n 5 e d. n i sin i as a definite integral on the given interval [,π]. i= 6. Use the form of the definition of the integral to evaluate the integral 5 (+3)d. 7. Use Part of the Fundamental Theorem of Calculus to find the derivative of the functions g() = lntdt and F() = cost dt. 5 π 8. Use Part of the Fundamental Theorem of Calculus to evaluate the integrals 5 3dand cos θdθ, π or eplain why it does not eist. 9. Find the derivative of the function. Find the general indefinite integral. 3 d ( t)(+t )dt ( ++ + )d sin sin d. Evaluate the integrals. g() = tan +t 4 dt. 5 π 4 3π (4y 3 + y 3)dy (e +4cos)d +cos cos d sin d
. The area of the region that lies to the right of the y-ais and to the left of the parabola = y y is given by the integral (y y )dy. (Turn your head clockwise and think of the region as lying below the curve = y y from y = to y =.) Find the area of the region. 3. The boundaries of the shaded region are the y-ais, the line y =, and the curve y = 4. Find the area of this region by writing as a function of y and integrating with respect to y. 4. Evaluate the indefinite integrals. +4 d ++ (ln) d e +e d cotcscd (e) + + d 5. Evaluate the definite integrals if eist. π 3 e 4 e a sin cos d d ln a d 6. Evaluate the integrals. (ln) d e sin3d cos(ln )d 7. First make a substitution and then use integration by parts to evaluate the integral π π 3 cos d. 8. Use integration by parts to prove the reduction formula (ln) n d = (ln) n n (ln) n d. 9. Evaluate the integrals. 3π 4 π π sin 5 cos 3 d cos d
(e) π sin cos d tan 3 secd sin5sind. Find the average value of the function f() = sin cos 3 on the interval [ π,π].. Find the area of the shaded region. = y 4y = y y. Sketch the region enclosed by the given curves. Decide whether to integrate with respect to or y. Draw a typical approimating rectangle and label its height and width. Then find the area of the region. = y,+y =. y = cos,y = sin, =, = π. 3. Evaluate the integrals. π π 4 cot 3 d. csc 4 cot 6 d. 4. Evaluate the integral the associated right triangle. 5. Evaluate the integrals. 5 d. 7 d. d ( ++). 3 d using the trigonometric substitution = 3tanθ. Sketch and label +9 3
6. Evaluate the integrals. +3 (+) d. ( )( 9) d. 3 ++ 4 +5 d. +4 7. Evaluate the integrals. e arctany +y dy. sin d. + d. e d. 8. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 3 e d. 4d. 9. Find the length of the curve y = +6 3,. 3. Find the area of the surface obtained by rotating the curve = 3 (y +) 3, y about the -ais. 3. Thecurve = acosh( y ), a y a, is rotated about the -ais. Find the area of the resulting surface. a 3. Estimate the area under the graph of f() = 5 from = to = 5 using five approimating rectangles and right endpoints. 33. Evaluate the integral θtan θdθ. 34. Sketch the region enclosed by the given curves y =,y = and = 9. Decide whether to integrate with respect to or y. Draw a typical approimating rectangle and label its height and width. Then find the area of the region. 35. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the curves y =,y = sin, π about the line y =. 36. Determine whether the integral integral. ln d is convergent or divergent. If it is convergent, evaluate the 37. Find a formula for the general term a n of the sequence, assuming that the pattern of the first few terms continues. {,7,,7,...} {, 3, 4 9, 8 7,...} 4
38. Determine whether the sequence converges or diverges. If it converges, find the limit. a n = 3+5n n+n a n = cos n a n = ln(n+) lnn a n = ( + n) n 39. Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a n = n+3 a n = n+ n 4. A sequence {a n } is given by a =, a n+ = +a n. By induction or otherwise, show that {a n } is increasing and bounded above by 3. Show that lim n a n eists. Find lim n a n. 4. Let a n = n 3n+. Determine whether {a n } is convergent or not. Determine whether a n is convergent or not. 4. Determine whether the series is convergent or divergent. n= ( 3) n 4 n, n, arctan n. 43. Find the values of for which the series of. n converges. find the sum of the series for those values 3n 44. Use the integral test to determine whether the series is convergent or divergent. 3n+, ne n, 3n+ n(n+), 5
n= nlnn. 45. Use the comparison test to determine whether the series is convergent or divergent. (e) 4+3 n n, n 3n 4 +, n +, n! n n, sin n. 46. Suppose that a n and b n are series with positive terms and b n is divergent. Prove that if a n lim = then a n is also divergent. n b n Use part to show that the series n= lnn and 47. Use alternating series test to determine whether the series convergent or not. lnn n are divergent. ( ) n+ n n 3 +4 and ( ) n sin π n are 48. Determine whether the following series is absolutely convergent, conditionally convergent, or divergent. ( ) n, n! n= ( ) n n n +, e n (n!). 6