MATH 101 HOMEWORK 3
Due on Wednesday Sept. 24
Covers sections 5.4–5.6. For full credit, please show all work.
1. (2 marks) We know that the average value of f (x) = x3 + 1 on some interval [a, b] is 9.
Prove that b ≥ 2. Can we say anything about a?
0
Z
sin x
2
ex
2. (3 marks) Find F (x), if F (x) =
+t2
dt.
0
3. (9 marks) Evaluate the integrals:
Z 3π/2
(a)
| sin5 x| dx,
0
Z
(b)
Z
(c)
1
2
x2 − x + 1
√
dx,
x
dx
.
ex + e−x
4. (6 marks) Evaluate the areas of the following planar regions:
(a) the region bounded from below by the lines y = x and y = −x, and from above by the
parabola y = 2 − x2 ,
(b) the region bounded by the half-parabola y = x2 , x ≥ 0; the half-parabola y = (x − 2)2 ,
x ≥ 2; and the horizontal lines y = a2 and y = b2 .
1