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 \( H \).

square pyramid used in problem

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.
projection of pyramid on x-y plane

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 .

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