Question

Find the volume of a trapezoid with vertices at (2,0), (2,2), (4,0), and (4,4); about the x-axis.

Answer 1

The trapezoid have vertices at (2,0), (2,2), (4,0), and (4,4)
and the rotation about the x-axis.
the points (2,0), (4,0) are located on the x-axis
calculate the function y = m x + c using the other vertices (2,2), (4,4)
by substituting in the function y
m = 1 , and  c =0
∴ y = x

The volume can be calculated using
 $V =\int\limits^a_b { \pi y^2 } \, dx$

∴ $V = \int\limits^4_2 { \pi y^2 } \, dx = \int\limits^4_2 { \pi x^2 } \, dx$
∴$V = \pi \ * \frac{x^3}{3}$
∴$V = \frac{ \pi }{3} * (4^3-2^3)= \frac{56 \pi }{3}$