Exercise 3.3

MA111: Prepared by Dr.Archara Pacheenburawana 43
(d) y = e x , y = x−1, −2 ≤ x ≤ 0
(e) y = x+1, y = (x−1) 2 , −1 ≤ x ≤ 2
(f) y = x 2 −1, y = 1−x, 0 ≤ x ≤ 2
(g) y = cosx, y = sin2x, 0 ≤ x ≤ π/2
(h) y = x 3 −1, y = 1−x, −2 ≤ x ≤ 2
3. Find the area of the region enclosed by the given curves.
(a) y = x 2 −1, y = 7−x 2
(b) y = x 2 +1, y = 3x−1
(c) y = x 3 , y = 3x+2
(d) y = x 3 , y = x 2
(e) y = x, y = 2−x, y = 0
(f) x = 3y, x = 2+y 2
(g) x = y, x = −y, x = 1
(h) y = x, y = 2, y = 6−x, y = 0
(i) x = 1−y 2 , x = y 2 −1
(j) y = cosx, y = 1−2x/π
(k) y = x 2 , y = 2/(x 2 +1)
(l) x = y 3 −y, x = 1−y 4
Answer to Exercise 5.1
1. (a) 6 (b) 40 3
(c) 9 2
2. (a) 20 (b) 40 (c) 19.5 (d) −e −2 (e) 31 (f) 3 (g) 1 (h) 29
3 3 6 2 2
3. (a) 64
3
(b) 1 6
(c) 27
4
(d) 1
12
(e) 1 (f) 1 6
(g) 1 (h) 8 (i) 1 8
(j) 2− π 2
(k) π − 2 3
(l) 8 5
Exercise 5.2
1. Findthevolumeofthesolidobtainedbyrotatingtheregionboundedbyy = 2−x, y =
0, and x = 0 (a) about the x-axis (b) about the line y = 3.
2. Find the volume of the solid obtained by rotating the region bounded by y = √ x, y =
2, and x = 0 (a) about the y-axis (b) about the line x = 4.
3. Find the volume of the solid obtained by rotating the region bounded by y = e x , x =
0, x = 2, and y = 0 (a) about the y-axis (b) about the line y = −2.