Notes: `x = sqrt(sin(y)), 0 |

Sunday, 21 May 2017

Let's use the method of cylindrical shells.

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

The radius of a cylinder (the distance to the axis of rotation) is $\displaystyle{4}-{y},$

the height of a cylinder is $\displaystyle\sqrt{{{\sin{{\left({y}\right)}}}}}.$

The volume is $\displaystyle{2}\pi{\int_{{0}}^{\pi}}{\left({4}-{y}\right)}\sqrt{{{\sin{{\left({y}\right)}}}}}{\left.{d}{y}\right.}.$

I believe "calculator" means "computer algebra system" here. WolframAlpha says the answer is 36.57476.

Let's use the method of cylindrical shells.

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

The radius of a cylinder (the distance to the axis of rotation) is $\displaystyle{4}-{y},$

the height of a cylinder is $\displaystyle\sqrt{{{\sin{{\left({y}\right)}}}}}.$

The volume is $\displaystyle{2}\pi{\int_{{0}}^{\pi}}{\left({4}-{y}\right)}\sqrt{{{\sin{{\left({y}\right)}}}}}{\left.{d}{y}\right.}.$

I believe "calculator" means "computer algebra system" here. WolframAlpha says the answer is 36.57476.

Notes