Review problems Ch 7
1.
Explain how to evaluate each integral – choose the appropriate technique
a. ∫
3𝑥
3
(𝑥 2 −9) ⁄2
dx
2
b. ∫ 𝑥 3 𝑒 −𝑥 𝑑𝑥
c.
d.
e.
3𝑥+2
𝑑𝑥
𝑥 2 +6𝑥+8
3𝑥+2
∫ (𝑥+2)(𝑥+1)(𝑥−1) 𝑑𝑥
5
∫ 𝑥 3 (2 + 𝑥 2 ) ⁄2 𝑑𝑥
2. Consider the region S bounded by y=x+(1/x2) and y=x-(1/x2) for x>1
a. Is the area finite or infinite? If finite find the area.
b. Suppose we rotate S around the x axis. Is the volume of the solid finite or infinite? If finite find the volume.
∞ 𝑥 2 +𝑥+2
𝑥 4 +𝑥 2 +1
3 𝑑𝑥
integral ∫2
3
(𝑥−2)2
3. Determine whether ∫2
4. Consider the
converges or diverges. Give reasons for your answer.
a. Why is this an improper integral?
b. Determine whether the integral converges or diverges and if it converges, compute its value.
5.
4
Consider the integral ∫0 √𝑥 2 + 1𝑑𝑥 , the endpoint approximations, L4, R4, the midpoint approximation M4, and
the trapezoidal approximation, T4
a. Which of the two approximations are likely to be more accurate in this case?
b. Which one of the two in part a is likely to be the most accurate?
c. Compute the approximation you chose in part b.
6. a) Show that the following formula is valid for any differentiable function f.
∫ 𝑥𝑓′(𝑥)𝑑𝑥 = 𝑥𝑓(𝑥) − ∫ 𝑓(𝑥)𝑑𝑥
b) Compute ∫ 3
𝑥
√𝑥−1
using the above formula
6
c) Suppose that f(x) is a continuous and differentiable, f(x)= -2, f(6) = -6 and ∫2 𝑓(𝑥)𝑑𝑥 = -10. Compute
6
∫2 𝑥𝑓 ′ (𝑥)𝑑𝑥
1
7. Let f(x) be the function graphed below. WE wish to approximate ∫0 𝑓(𝑥)𝑑𝑥 Which of the following gives the best
approximation and why?
A) MIDPOINT RULE WITH 2 SUBINTERVALS
B) Midpoint rule with 4 subintervals
c) Left end point rule with 4 subintervals.
8. Let f(x) be the function graphed below.
Four students approximated the area under the curve and each used a different method, but they all used the same number of
subintervals. Here are their results:
g.
V.
T.
P
2.458
2.638
2.555
2.178
Which student used which method? Explain.
9. Consider the three functions graphed below
Which of the following must be true, cannot be true or might be true?
∞
a)∫1 𝑓(𝑥)𝑑𝑥 converges
1
b) ∫0 𝑔(𝑥)𝑑𝑥 converges
∞
c) ∫1 𝑔(𝑥)𝑑𝑥 diverges
10. Integrate the following:
∫ √1 + 𝑒 𝑥 𝑑𝑥
𝑠 3 +1
∫ 𝑠3−𝑠2 ds
𝑥𝑒 𝑥
∫ √1+𝑒 𝑥 𝑑𝑥