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Exercise 3.3

Exercise 3.3

Exercise 3.3

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MA111: Prepared by Dr.Archara Pacheenburawana 44<br />

4. Findthevolumeofthesolidobtainedbyrotatingtheregionboundedbyy = x 3 , y = 0,<br />

and x = 1 (a) about the y-axis (b) about the x-axis.<br />

5. The region R enclosed by the curves y = 3−x, the x-axis, and the y-axis. Find the<br />

volume of the solid obtained by rotating the region R (a) about the y-axis, (b) about<br />

the x-axis (c) about the line y = 3, (d) about the line y = −3, (e) about the line<br />

x = 3, and (f) about the line x = −3.<br />

6. The region R enclosed by the curves y = x 2 , y = 0, and x = 1. Find the volume of<br />

the solid obtained by rotating the region R (a) about the y-axis, (b) about the x-axis,<br />

(c) about the line x = 1, (d) about the line y = 1, (e) about the line x = −1, and (d)<br />

about the line y = −1.<br />

1. (a) 8π 3<br />

4. (a) 2π 5<br />

(b) 28π<br />

3<br />

(b) π 7<br />

Answer to <strong>Exercise</strong> 5.2<br />

2. (a) 32π<br />

5<br />

(b) 224π<br />

15<br />

( e<br />

4<br />

3. (a) 2πe 2 +2π (b) π<br />

2 +4e2 − 9 )<br />

2<br />

5. (a) 9π (b) 9π (c) 18π (d) 36π (e) 18π (f) 36π<br />

6. (a) π 2<br />

(b) π 5<br />

(c) π 6<br />

(d) 7π<br />

15<br />

(e) 7π 6<br />

(f) 13π<br />

15<br />

<strong>Exercise</strong> 5.3<br />

1. Use the method of cylindrical shells to find the volume generated by rotating the<br />

region bounded by the given curve about the y-axis.<br />

(a) y = 1 , y = 0, x = 1, x = 2<br />

x<br />

(b) y = e −x2 , y = 0, x = 0, x = 1<br />

(c) y 2 = x, x = 2y<br />

2. Usethemethodofcylindricalshellstofindthevolumeofthesolidobtainedbyrotating<br />

the region bounded by the given curve about the x-axis.<br />

(a) x = 1+y 2 , x = 0, y = 1, y = 2<br />

(b) y = x 2 , y = 9<br />

(c) y = √ x, y = 0, x+y = 2<br />

3. Use the appropriate method to find the volume generated by rotating the region<br />

bounded by the given curve about the specified axis.<br />

(a) y = √ x, x = 3, y = 0; about y-axis<br />

(b) y = √ x, x = 5, y = 0; about x = 5<br />

(c) y = 1 4 x3 +1, y = 1−x, x = 1; about y-axis<br />

(d) x = y 2 , y = 1, x = 0; about x-axis<br />

(e) x = y 2 , y = 2, x = 0; about y = 2

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