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6.2. USING DEFINITE INTEGRALS TO FIND VOLUME
Activity 6.5.
In each of the following questions, draw a careful, labeled sketch of the region described,
as well as the resulting solid that results from revolving the region about the stated axis.
In addition, draw a representative slice and state the volume of that slice, along with a
definite integral whose value is the volume of the entire solid. It is not necessary to
evaluate the integrals you find.
(a) The region S bounded by the y-axis, the curve y =
√
x, and the line y = 2;
revolve S about the y-axis.
(b) The region S bounded by the x-axis, the curve y =
√
x, and the line x = 4;
revolve S about the y-axis.
(c) The finite region S in the first quadrant bounded by the curves y = 2x and
y = x 3 ; revolve S about the x-axis.
(d) The finite region S in the first quadrant bounded by the curves y = 2x and
y = x 3 ; revolve S about the y-axis.
(e) The finite region S bounded by the curves x = (y − 1) 2 and y = x − 1; revolve
S about the y-axis
⊳
Revolving about horizontal and vertical lines other than the coordinate axes
Just as we can revolve about one of the coordinate axes (y = 0 or x = 0), it is also possible
to revolve around any horizontal or vertical line. Doing so essentially adjusts the radii of
cylinders or washers involved by a constant value. A careful, well-labeled plot of the solid
of revolution will usually reveal how the different axis of revolution affects the definite
integral we set up. Again, an example is instructive.
Example 6.4. Find the volume of the solid of revolution generated when the finite region
S that lies between y = x 2 and y = x is revolved about the line y = −1.
Solution.
Graphing the region between the two curves in the first quadrant between their points
of intersection ((0, 0) and (1, 1)) and then revolving the region about the line y = −1, we
see the solid shown in Figure 6.11. Each slice of the solid perpendicular to the axis of
revolution is a washer, and the radii of each washer are governed by the curves y = x 2
and y = x. But we also see that there is one added change: the axis of revolution adds a
fixed length to each radius. In particular, the inner radius of a typical slice, r(x), is given
6.2. USING DEFINITE INTEGRALS TO FIND VOLUME
Activity 6.5.
In each of the following questions, draw a careful, labeled sketch of the region described,
as well as the resulting solid that results from revolving the region about the stated axis.
In addition, draw a representative slice and state the volume of that slice, along with a
definite integral whose value is the volume of the entire solid. It is not necessary to
evaluate the integrals you find.
(a) The region S bounded by the y-axis, the curve y =
√
x, and the line y = 2;
revolve S about the y-axis.
(b) The region S bounded by the x-axis, the curve y =
√
x, and the line x = 4;
revolve S about the y-axis.
(c) The finite region S in the first quadrant bounded by the curves y = 2x and
y = x 3 ; revolve S about the x-axis.
(d) The finite region S in the first quadrant bounded by the curves y = 2x and
y = x 3 ; revolve S about the y-axis.
(e) The finite region S bounded by the curves x = (y − 1) 2 and y = x − 1; revolve
S about the y-axis
⊳
Revolving about horizontal and vertical lines other than the coordinate axes
Just as we can revolve about one of the coordinate axes (y = 0 or x = 0), it is also possible
to revolve around any horizontal or vertical line. Doing so essentially adjusts the radii of
cylinders or washers involved by a constant value. A careful, well-labeled plot of the solid
of revolution will usually reveal how the different axis of revolution affects the definite
integral we set up. Again, an example is instructive.
Example 6.4. Find the volume of the solid of revolution generated when the finite region
S that lies between y = x 2 and y = x is revolved about the line y = −1.
Solution.
Graphing the region between the two curves in the first quadrant between their points
of intersection ((0, 0) and (1, 1)) and then revolving the region about the line y = −1, we
see the solid shown in Figure 6.11. Each slice of the solid perpendicular to the axis of
revolution is a washer, and the radii of each washer are governed by the curves y = x 2
and y = x. But we also see that there is one added change: the axis of revolution adds a
fixed length to each radius. In particular, the inner radius of a typical slice, r(x), is given