Question

Anyone who knows how to do this? (Volume of revolution)

A volume is described as follows:
1. the base is the region bounded by x = - y^2+4 y+79 and x= y^2-22 y+123;
2. every cross section perpendicular to the y-axis is a semi-circle.

Find the volume of this object.

Answer 1

Apologies, I had to stare at this one blankly for a while to recall the methods.
Anyways, what have you tried so far?

Answer 2

One thing that I do not like in the problem is those functions defined x in terms of y as sideways parabolas. You might be able to swap the variables x and y along with the perpendicular axis to the cross sections' and make the problem generally easier to manage.

Answer 3

\(x_1 = - y^2+4 y+79 \\ x_2= y^2-22 y+123\)

u may get the volume element as :
\(\large dV = \frac{\pi (\frac{x_2 - x_1}{2})^2}{2} dx \)

integrating that over 2->11 would give the volume of required solid

Answer 4

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\(\large dV = \frac{\pi (\frac{x_2 - x_1}{2})^2}{2}\color{red}{ dy} \)