# Volume of a Solid with a Known Cross Section

In this topic, we will learn how to find the volume of a solid object that has known cross sections.

We consider solids whose cross sections are common shapes such as triangles, squares, rectangles, trapezoids, and semicircles.

## Definition: Volume of a Solid Using Integration

Let *S* be a solid and suppose that the area of the cross section in the plane perpendicular to the *x*-axis is *A* (*x*) for *a* ≤ *x* ≤ *b*.

Then the volume of the solid from \(x = a\) to \(x = b\) is given by the cross-section formula

Similarly, if the cross section is perpendicular to the \(y-\)axis and its area is defined by the function \(A\left( y \right),\) then the volume of the solid from \(y = c\) to \(y = d\) is given by

## Steps for Finding the Volume of a Solid with a Known Cross Section

- Sketch the base of the solid and a typical cross section.
- Express the area of the cross section \(A\left( x \right)\) as a function of \(x.\)
- Determine the limits of integration.
- Evaluate the definite integral
\[V = \int\limits_a^b {A\left( x \right)dx}.\]

## Solved Problems

### Example 1.

The solid has a base lying in the first quadrant of the \(xy-\)plane and bounded by the lines \(y = x,\) \(x = 1,\) \(y = 0.\) Every planar section perpendicular to the \(x-\)axis is a semicircle. Find the volume of the solid.

Solution.

The diameter of the semicircle at a point \(x\) is \(d=y=x.\) Hence, the area of the cross section is

Integration yields the following result:

### Example 2.

Find the volume of a solid bounded by the elliptic paraboloid \[z = \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}}\] and the plane \(z = 1.\)

Solution.

Consider an arbitrary planar section perpendicular to the \(z-\)axis at a point \(z,\) where \(0 \lt z \le 1.\) The cross section is an ellipse defined by the equation

The area of the cross section is

Then, by the cross-section formula,

### Example 3.

The base of a solid is bounded by the parabola \[y = 1 - {x^2}\] and the \(x-\)axis. Find the volume of the solid if the cross sections are equilateral triangles perpendicular to the \(x-\)axis.

Solution.

The area of the equilateral triangle at a point \(x\) is given by

As the side \(a\) is equal to \(1-{x^2},\) then

The parabola \(y = 1 - {x^2}\) intersects the \(x-\)axis at the points \(x=-1,\) \(x = 1.\)

Compute the volume of the solid:

### Example 4.

Find the volume of a regular square pyramid with the base side \(a\) and the altitude \(H.\)

Solution.

The area of the square cross section at a point \(x\) is written in the form

Hence, the volume of the pyramid is given by