Homework 4Solutions

Exercise 1. Determine whether each integral is convergent or divergent. Evaluate thosethat are convergent.

1.

∫ ∞1

1

(3x+ 1)2dx

2.

∫ −1−∞

1√2− w

dw

3.

∫ ∞2π

sin θ dθ

4.

∫ 1

0

3

x5dx

1

5.

∫ 14

−2

dx4√x+ 2

dx

Exercise 2. Sketch the region enclosed by the given curves. Decide whether to integratewith respect to x or y, and find the area of the region.

1. y = x+ 1, y = 9− x2, x = −1, x = 2

2. y = sinx, y = ex, x = 0, x = π/2

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3. y = 1 +√x, y = 1 + 1

3

4. y = x2, y = 4x− x2

3

Exercise 3. Find the volume of the solid obtained by rotating the region bounded bythe given curves about the specified line using an appropriate method (disks, washers, orcylinders). Sketch the region, the solid, and a typical disk, washer, or cylinder.

1. y = lnx, y = 1, y = 2, x = 0; about the y − axis

2. y2 = x, x = 2y; about the y − axis

3. y = x2, y = 0, x = 1, x = 2; about x = 1

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4. y = x, y =√x; about y = 1

5. y = x2, x = y2, about y = −1

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