The base of a solid is the region in the first quadrant bounded by the line x + 2y = 4 and the coordinate axes. What is the volume of the solid if every cross section perpendicular to the x-axis is a semicircle?

Solution:

Given, the volume of the solid represents a semi circle.

Area of semi circle is given by πr²/2

The cross section area of the solid at point x, is given by

A(x) = πy²/8 where y = r/2.

The volume of the solid is

\(\\V=\int_{0}^{4}A(x) dx \\ \\=\int_{0}^{4}\frac{\pi y^{2}}{8} dx \\ \\=\frac{\pi }{8}\int_{0}^{4}[\frac{(4-x)}{2}]^{2}dx \\ \\=\frac{\pi }{32}\int_{0}^{4}(4-x)^{2}dx\)

= (π/32)(64/3)

= 2π/3

Therefore, the volume of the solid is 2π/3.

The base of a solid is the region in the first quadrant bounded by the line x + 2y = 4 and the coordinate axes. What is the volume of the solid if every cross section perpendicular to the x-axis is a semicircle?

Summary:

If the base of a solid is the region in the first quadrant bounded by the line x + 2y = 4, then the volume of the solid if every cross section perpendicular to the x-axis is a semicircle is 2π/3.