Question

Let R be the region bounded by the x-axis, x = 5 and the curve
y = x. This region is rotated around the x-axis. Find the volume of the resulting
solid. (Note: R is a triangular region. The resulting solid has a simple shape.
You may ask me if you are highly unsure about what the solid "looks like.")

Answer 1

geometrically, the solid is a cone of height=5, radius=5

v = 125/3 pi

Analytically,

v = Integral(pi * y^2 dx)[0,5]
= Integral(pi * x^2 dx)[0,5]
= pi/3 x^3 [0,5]
= 125/3 pi