Calculate Volume of a Solid with Known Cross Section using Integration: Math Tutorial

volume of known cross sections

v(t) = 0

The volume of a solid with a known cross section can be found by using integration. The idea is to slice the solid into small pieces that have the known cross section and then to add up the volumes of these pieces using integration.

Let’s consider an example:

Suppose we have a solid which is bounded by the x-axis, the line y = x, and the line y = x^2. We want to find the volume of this solid with a known cross section, which is a rectangle whose height is 2.

To find the volume, we have to consider a slice of thickness dx. The area of the cross section is the height times the width of the rectangle, which is 2x – x = x. So, the volume of this slice is:

dV = (2 * x) * dx

To find the total volume, we need to add up all the slices from x = 0 to x = 1. So, we integrate dV over this range:

V = ∫₀¹ (2 * x) dx

V = [x²]₀¹ = 1

Therefore, the volume of the solid is 1 cubic unit.

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