- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
13.25. Miscellaneous review exercises on conic sections
- The region bounded by the parabola $y^{2} = 8x$ and the line $x = 2$ is rotated about the $x$ axis. Find the volume of the solid of revolution so generated.
- The volume of a solid of revolution with cross sections cut along the $x$ axis is $$\int_{a}^{b}\pi f^{2}(x) \ dx$$ But if we define $y = f(x),$ then $f^{2}(x) = y^{2} = 8x.$ We know from the parabola's equation that its vertex is $(0, 0),$ and since the cross sectional region is bound by the vertical line at $x = 2,$ our limits of integration must be $a = 0$ and $b = 2.$
- The solid of revolution generated by rotating the above defined cross-sectional area around the $x$ axis is: $$\displaylines{V = 8\pi \int_{0}^{2}x \ dx \\ = 4\pi x^{2}\Biggr|_{0}^{2} \\ = 16\pi \ \blacksquare}$$