§5.1 Homework Problems
Directions: For each problem, complete the following steps:
1 Sketch the region bounded by the given curves. (Remember that you’ll need the intersection
points to do this.) Label each curve appropriately with either yT , yB , xR or xL .
2 Draw a representative rectangle (horizontally or vertically), and label its height and width.
3 Set up the integral by approximating with An and then determining A(R) = lim An .
n→∞
4 Calculate A(R), as defined by the integral in step 3.
The answers appear next to (most) problems to help you.
1
1. (from class) Do only step 4 for the region bounded by x = y + 1 and x = y 2 − 3.
2
34
2. y = 5x − x2 , y = 0, x = 1, x = 3. Ans:
3
81
3
3. y = x , y = 0, x = −3, x = 3. Ans:
2
8
4. x = 1 − y 2 , x = y 2 − 1. Ans:
3
π2
π
−1
5. y = sin x, y = x, x = 0, x = . Ans:
2
8
1
6. y = x, y = x2 . Ans:
6
3
7. y = x − x, y = 3x for x ≥ 0 only. Ans: 4
9
8. x + y 2 = 2, x + y = 0. Ans:
2
20
9. y = |x|, y = x2 − 2. Ans:
3
8
10. x = y 3 − y, y 4 = 1 − x. Ans:
5