For each problem, use the method of cylindrical shells to find the volume of the
solid that results ... π The cylindrical shell method requires one integral, while the.
Kuta Software - Infinite Calculus
Name___________________________________
Volumes by Cylindrical Shells
Date________________ Period____
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the y-axis. 2) y = 7 y= x x=0 x=4
1) y = x + 4 2 y=x +4 y 8 6
y 8
4 6 2 4 −8
−6
−4
−2
2
4
6
8 x
2
−2 −8
−4
−6
−4
−2
2
4
6
8 x
−2 −6 −4 −8 −6 −8
Critical thinking question: 3) Solve problem 2 using the method of washers. Why is this problem easier using cylindrical shells?
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the y-axis. You may use the provided graph to sketch the curves and shade the enclosed region. 4) y = 2 x 2 y=x y 8 6 4 2 −8
−6
−4
−2
2
4
6
8 x
−2 −4 −6 −8
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the given axis. You may use the provided graph to sketch the curves and shade the enclosed region. 2
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the y-axis. 2) y = 7 y= x x=0 x=4
1) y = x + 4 2 y=x +4 y 8 6
y 8
4 6 2 4 −8
−6
−4
−2
2
4
6
8 x
2
−2 −8
−4
−6
−4
−2
2
4
6
8 x
−2 −6 −4 −8 −6
2π
∫
−8
1
x(
x + 4 − ( x 2 + 4)) dx
0
3 π 10
=
2π
∫
4
x(7 −
x ) dx
0
432 = π 5
Critical thinking question: 3) Solve problem 2 using the method of washers. Why is this problem easier using cylindrical shells? π
∫
2
0
( y 2 ) 2 dy + π
∫
7 2
4 dy =
2
432 π The cylindrical shell method requires one integral, while the 5
For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the y-axis. You may use the provided graph to sketch the curves and shade the enclosed region. 4) y = 2 x 2 y=x y 8 6 4 2 −8
−6
−4
−2
2
4
6
8 x
−2 −4 −6 −8
2π
∫
2
x(2 x − x 2 ) dx
0
8 = π 3 For each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the the given axis. You may use the provided graph to sketch the curves and shade the enclosed region. 2
5) y = − x + 7 2 y=x +5 Axis: x = 2 y 8 6 4 2 −8
−6
−4
−2
2
4
6
8 x
−2 −4 −6 −8
2π
∫
1
(2 − x)(− x 2 + 7 − ( x 2 + 5)) dx
−1
32 = π 3 Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com
