Volume of Solids with a Known Cross Section

A solid of revolution is formed by revolving a region in the xy-plane about a line. The line about which a planar region is revolved is the axis of revolution. It is often necessary to find the volume of a solid of revolution, which refers to the amount of space occupied by a planar region that has been revolved about an axis. In some cases the volume of a solid can be determined using geometric formulas, such as those for the volume of a cube, sphere, cylinder, or pyramid. However, in most cases the solids for which volumes are needed do not have explicit formulas and calculus is needed.

Solids with Known Cross Sections

image of baguette Imagine a loaf of unsliced bread lying along the x-axis in the xy-plane between x = a and x = b. The typical way to slice a loaf of bread is by making cuts perpendicular to the x-axis. Each slice is considered a cross section. In mathematical terms a cross section is a plane region formed by slicing (cutting) through an object at an angle perpendicular to its axis. For any value of x between a and b, let A(x) represent the area of a cross section of the loaf of bread generated when a knife slices through the bread perpendicular to the x-axis. Recognizing that A(x) varies with x as it rises and falls with the loaf's thickness, assume that each slice is a right circular cylinder. Then each cross-sectional area is equal to equation image indicator $LaTeX: \pi r^2$ and the thickness of the slice of bread is h. The volume of each slice can be determined by using the volume formula for a right circular cylinder $LaTeX: \left(V=\pi r^2h\right)$ . In terms of cross-sectional area and thickness, the volume of one slice is equation image indicator $LaTeX: A\left(x\right)\Delta x$ . The total volume is then approximated by summing the volumes of each cylinder as the number of cylinders grows toward infinity, which is represented as $LaTeX: V=\lim_{x \to \infty }\ \sum_{i=1}^{n} A\left(x\right)\Lambda x$ . This limiting sum is written as equation image indicator $LaTeX: A=\int_a^bArea\left(x\right)dx\:or\:V=\int_a^b\pi\left(radius\right)^2dx$ .

View the following video below which uses squares to find the volume of the solid.

View the following video which uses right triangles to find the volume of the solid.

Volume of Solids with a Known Cross Section

1. Find the volume of the solid whose base is bounded by the circle equation image indicator LaTeX: x^2+y^2=4 if the cross sections perpendicular to the x-axis are squares.

graph to example 1 (circle) Solution: text annotation indicator The area of the square is given by A=s². The equation for the circle can be rewritten from x²+y²=4 as y²=4-x or y=√(4-x²). One side of the square, perpendicular to the x-axis, is given by s=2√(4-x²), as we account for the portion of the circle above and below the x-axis (symmetric, so we double the equation.) The volume can be found by:

$LaTeX: V=\int_{-2}^2\left(2\sqrt[]{4-x^2}\right)^{^2}dx\\ \int_{-2}^24\left(4-x^2\right)dx\\ \int_{-2}^216-4x^2dx=\frac{128}{3}$

2. Find the volume of the solid whose base is bounded by y = x + 1 and equation image indicator LaTeX: y=x^2-1 if the cross sections perpendicular to the x-axis are rectangles of height 5.

graph to example 2 (parabola opening up with line) Solution: text annotation indicator The graphs intersect at (-1,0) and (2,0). The area of the rectangle can be given by A=bh, and we are given the height of each rectangle is 5. To find the base, we subtract the lower curve from the upper curve (x+1)-(x²-1), or -x²+x+2. Thus our volume can be found by

$LaTeX: V=\int_{-1}^{2} bhdx\\ \int_{-1}^{2}\left(-x^2+x+2\right)\left(5\right)dx=22.5$

Volume of Solids with a Known Cross Section: Even More Problems

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of the volumes of solids with a known cross section.

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