Find The Volume of a Square Pyramid Using Integrals
Find the formula for the volume of a square pyramid using integrals in calculus.
Problem : A pyramid is shown in the figure below. Its base is a square of side \( a \) and is orthogonal to the y axis. The height of the pyramid is \( H \). Use integrals and their properties to find the volume of the square pyramid in terms of \( a \) and \( H \).
Solution to the problem:
Let us first position the pyramid so that two opposite sides of the square base are perpendicular to the x axis and the center of its base is at the origin of the x-y system of axes. If we look at the pyramid in a direction orthogonal to the x-y plane, it will look like a two dimensional shape as shown below. AC is the slant height.
Let \( x = A'B' \) be the length of half of the side of the square at height \( y \). The area \( A \) of the square at height \( y \) is given by:
\[ A(x) = (2x)^2 \]
The volume is found by adding all the volumes \( A \, dy \) that make the pyramid from \( y = 0 \) to \( y = H \). Hence
Volume = \( \int_{0}^{H} A^2 \, dy \)
\[ = 4 \int_{0}^{H} x^2 \, dy \]
We now use the fact that triangles ABC and AB'C' are similar and therefore the lengths of their corresponding sides are proportional to write:
\[ \frac{a/2}{x} = \frac{H}{H - y} \]
We now solve the above for \( x \) to obtain
\[ x = \frac{a (H - y)}{2 H} \]
We now substitute \( x \) in the integral that gives the volume to obtain
\[ \text{Volume} = 4 \left(\frac{a}{2H}\right)^2 \int_{0}^{H} (H - y)^2 \, dy \]
Let us define \( t \) by
\[ t = H - y \quad \text{and} \quad dt = - dy \]
Substitute and change the limits of integration to write the volume as follows:
\[ \text{Volume} = 4 \left(\frac{a}{2H}\right)^2 \int_{H}^{0} t^2 (- dt) \]
Evaluate the integral and simplify
\[ \text{Volume} = 4 \left(\frac{a}{2H}\right)^2 \left[\frac{H^3}{3}\right] \]
\[ \text{Volume} = \frac{a^2 H}{3} \]
The volume of a square pyramid is given by the area of the base times the third of the height of the pyramid.
More references and Links
integrals and their applications in calculus.Area under a curve .
Area between two curves .
Find The Volume of a Solid of Revolution .
Volume by Cylindrical Shells Method .