Notes: `y = tan(x), y = 0, x = pi/4` (a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about...

Wednesday, 21 December 2016

Let's use the method of cylindrical shells.

The parameter for a cylinder will be $\displaystyle{x}$ from $\displaystyle{x}={0}$ to $\displaystyle{x}=\frac{\pi}{{4}}.$

The radius of a cylinder (the distance to the axis) is $\displaystyle\frac{\pi}{{2}}-{x},$ the height is $\displaystyle{\tan{{\left({x}\right)}}}.$

The volume is $\displaystyle{2}\pi{\int_{{0}}^{{\frac{\pi}{{4}}}}}{\left(\frac{\pi}{{2}}-{x}\right)}{\tan{{x}}}{\left.{d}{x}\right.}.$

I believe "calculator" means "computer algebra system here". WolframAlpha says the answer is 2.25323.

Let's use the method of cylindrical shells.

The parameter for a cylinder will be $\displaystyle{x}$ from $\displaystyle{x}={0}$ to $\displaystyle{x}=\frac{\pi}{{4}}.$

The radius of a cylinder (the distance to the axis) is $\displaystyle\frac{\pi}{{2}}-{x},$ the height is $\displaystyle{\tan{{\left({x}\right)}}}.$

The volume is $\displaystyle{2}\pi{\int_{{0}}^{{\frac{\pi}{{4}}}}}{\left(\frac{\pi}{{2}}-{x}\right)}{\tan{{x}}}{\left.{d}{x}\right.}.$

I believe "calculator" means "computer algebra system here". WolframAlpha says the answer is 2.25323.

Notes