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6.2. USING DEFINITE INTEGRALS TO FIND VOLUME
In this section, we encountered the following important ideas:
• We can use a definite integral to find the volume of a three-dimensional solid of
revolution that results from revolving a two-dimensional region about a particular axis
by taking slices perpendicular to the axis of revolution which will then be circular disks
or washers.
• If we revolve about a vertical line and slice perpendicular to that line, then our slices
are horizontal and of thickness △y. This leads us to integrate with respect to y, as
opposed to with respect to x when we slice a solid vertically.
• If we revolve about a line other than the x- or y-axis, we need to carefully account for
the shift that occurs in the radius of a typical slice. Normally, this shift involves taking a
sum or difference of the function along with the constant connected to the equation for
the horizontal or vertical line; a well-labeled diagram is usually the best way to decide
the new expression for the radius.
Exercises
1. Consider the curve f (x) = 3 cos(
x 3
4 ) and the portion of its graph that lies in the first
quadrant between the y-axis and the first positive value of x for which f (x) = 0. Let R
denote the region bounded by this portion of f , the x-axis, and the y-axis.
(a) Set up a definite integral whose value is the exact arc length of f that lies along
the upper boundary of R. Use technology appropriately to evaluate the integral
you find.
(b) Set up a definite integral whose value is the exact area of R. Use technology
appropriately to evaluate the integral you find.
(c) Suppose that the region R is revolved around the x-axis. Set up a definite
integral whose value is the exact volume of the solid of revolution that is
generated. Use technology appropriately to evaluate the integral you find.
(d) Suppose instead that R is revolved around the y-axis. If possible, set up an
integral expression whose value is the exact volume of the solid of revolution
and evaluate the integral using appropriate technology. If not possible, explain
why.
2. Consider the curves given by y = sin(x) and y = cos(x). For each of the following
problems, you should include a sketch of the region/solid being considered, as well as a
labeled representative slice.
(a) Sketch the region R bounded by the y-axis and the curves y = sin(x) and
y = cos(x) up to the first positive value of x at which they intersect. What is
the exact intersection point of the curves?
6.2. USING DEFINITE INTEGRALS TO FIND VOLUME
In this section, we encountered the following important ideas:
• We can use a definite integral to find the volume of a three-dimensional solid of
revolution that results from revolving a two-dimensional region about a particular axis
by taking slices perpendicular to the axis of revolution which will then be circular disks
or washers.
• If we revolve about a vertical line and slice perpendicular to that line, then our slices
are horizontal and of thickness △y. This leads us to integrate with respect to y, as
opposed to with respect to x when we slice a solid vertically.
• If we revolve about a line other than the x- or y-axis, we need to carefully account for
the shift that occurs in the radius of a typical slice. Normally, this shift involves taking a
sum or difference of the function along with the constant connected to the equation for
the horizontal or vertical line; a well-labeled diagram is usually the best way to decide
the new expression for the radius.
Exercises
1. Consider the curve f (x) = 3 cos(
x 3
4 ) and the portion of its graph that lies in the first
quadrant between the y-axis and the first positive value of x for which f (x) = 0. Let R
denote the region bounded by this portion of f , the x-axis, and the y-axis.
(a) Set up a definite integral whose value is the exact arc length of f that lies along
the upper boundary of R. Use technology appropriately to evaluate the integral
you find.
(b) Set up a definite integral whose value is the exact area of R. Use technology
appropriately to evaluate the integral you find.
(c) Suppose that the region R is revolved around the x-axis. Set up a definite
integral whose value is the exact volume of the solid of revolution that is
generated. Use technology appropriately to evaluate the integral you find.
(d) Suppose instead that R is revolved around the y-axis. If possible, set up an
integral expression whose value is the exact volume of the solid of revolution
and evaluate the integral using appropriate technology. If not possible, explain
why.
2. Consider the curves given by y = sin(x) and y = cos(x). For each of the following
problems, you should include a sketch of the region/solid being considered, as well as a
labeled representative slice.
(a) Sketch the region R bounded by the y-axis and the curves y = sin(x) and
y = cos(x) up to the first positive value of x at which they intersect. What is
the exact intersection point of the curves?