Volume by Perpendicular Cross SectionsLesson Objective - after completing this lesson you will be able to calculate the volume of solids formed by slices of known shapes perpendicular to either the x-axis or y-axis.
We have seen that integration is used to compute the area of two-dimensional regions bounded by curves. Integrals are also used to find the volume of three-dimensional regions (or solids). Once again, the slice-and-sum method is the key to solving these problems. Consider a solid object that extends in the x-direction from x = a to x = b. Imagine making a vertical cut through the solid, perpendicular to the x-axis at a particular point x, and suppose the area of the cross section created by the cut is given by a known integrable function A. (For more information, visit Khan Academy) |
In Geometry, you learned formulas for finding volumes of common three-dimensional solids (cubes, spheres, cones, rectangular prisms, and perhaps others).
Calculus allows us to find volumes of solids whose bases are two-dimensional regions within an x-y coordinate system, and whose heights are formed by cross sections (most often squares, rectangles, semicircles, or triangles) which essentially “stick out from the base” to form the third dimension of the object.
Calculus allows us to find volumes of solids whose bases are two-dimensional regions within an x-y coordinate system, and whose heights are formed by cross sections (most often squares, rectangles, semicircles, or triangles) which essentially “stick out from the base” to form the third dimension of the object.
Example 2
Set up (but do not integrate) integrals for the volumes of the solids with the same base as in Example 1, but whose cross sections are:
a) Semicircles perpendicular to the x-axis. b) Rectangles of height 1/4 which are perpendicular to the y-axis.
In this problem we need the radius of the In this problem we once again need the length of the rectangle
semicircle. The radius is half the length The height of each rectangular slice is 1/4. Multiply length by
of the rectangle. width and you have the area.
Set up (but do not integrate) integrals for the volumes of the solids with the same base as in Example 1, but whose cross sections are:
a) Semicircles perpendicular to the x-axis. b) Rectangles of height 1/4 which are perpendicular to the y-axis.
In this problem we need the radius of the In this problem we once again need the length of the rectangle
semicircle. The radius is half the length The height of each rectangular slice is 1/4. Multiply length by
of the rectangle. width and you have the area.
Example 4
Set up integrals for the volumes of the solids whose base is circle x^2 + y^2 = 1 and whose cross sections are a) equilateral triangles and b) rectangles whose heights are three times their bases. For both problems we will use the length of the rectangle. To get this length, we will need to solve the equation for x. This is show below. This gives us half the length. a) We will use the formula for the area of an equilateral b) Rectangles whose heights are triangle. To find the length of the rectangle we three times their bases and subtract the right side minus the left side. whose bases are perpendicular to the y-axis. |
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This website was prepared by Brant Breeding in his personal capacity. These materials are not endorsed, approved, sponsored, or provided by or on behalf of the University of Arkansas, Fayetteville.
Images on this page courtesy of Briggs, Gillet, and Schulz and Adobe Stock website (full citation found on Reference page).
Please follow this link for our full disclaimer statement.