Calculus2

Regions Between Curves

� 1, 2 Find the area of the shaded region using (a) vertical strips, and (b) horizontal strips.

1.

x

y

1

y = ex

2.

x = y2

y = 2 − x

x

y

� 3–5 Scaling appropriately, (a) sketch the region enclosed by the given curves, and (b) compute its area using the method of your choice—vertical or horizontal strips.

3. y = sin x, y = ex, x = 0, x = π/2

4. x = y2 − 1, x = 1 − y2

5. y = 1/x, y = x, y = 1 4 x, x > 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. See Figure 1.

a) Showing your work, find the relevant points of intersection.

Hint: Use a trigonometric identity.

b) Find the area of the shaded region.

7. See Figure 2.

a) Find the equation of the tangent line.

b) Find the area of the shaded region. Hint: Use horizontal strips.

x

y

−1

1

y = sin 2x

y = sin x

Figure 1.

x

y

• (1, 1) y = x2

Tangent Line

Figure 2.

1 of 2

8. Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions of equal area.

Solutions to Selected Problems

1. a) e − 1

b) e − 1

2. a) 9

2

b) 9

2

3. a) Graph

b) e π/2

− 2

4. a) Graph

b) 8

3

5. a) Graph

b) ln 2

6. a) x = π

3 and π

b) 9

4

7. a) y = 2x − 1

b) 1

12

8. b = 2 4/3

2 of 2