Homework 4
Solutions
Exercise 1. Determine whether each integral is convergent or divergent. Evaluate those
that are convergent.
Z ∞
1
1.
dx
(3x + 1)2
1
Z
−1
√
2.
−∞
Z
1
dw
2−w
∞
3.
sin θ dθ
2π
Z
4.
0
1
3
dx
x5
1
Z
14
5.
−2
√
4
dx
dx
x+2
Exercise 2. Sketch the region enclosed by the given curves. Decide whether to integrate
with respect to x or y, and find the area of the region.
1. y = x + 1,
2. y = sin x,
y = 9 − x2 ,
y = ex ,
x = −1,
x = 0,
x=2
x = π/2
2
3. y = 1 +
4. y = x2 ,
√
x,
y =1+
1
3
y = 4x − x2
3
Exercise 3. Find the volume of the solid obtained by rotating the region bounded by
the given curves about the specified line using an appropriate method (disks, washers, or
cylinders). Sketch the region, the solid, and a typical disk, washer, or cylinder.
1. y = ln x,
y = 1,
2. y 2 = x,
x = 2y;
3. y = x2 ,
y = 0,
y = 2,
x = 0;
about the y − axis
about the y − axis
x = 1,
x = 2;
about x = 1
4
4. y = x,
5. y = x2 ,
y=
√
x;
about y = 1
x = y 2 , about y = −1
5