Question on Finding Volume of the Solid, Using Shell Method for setup and Evaluating the integral

Here is a Post on Finding Volume of a Solid by shell Method set up and Evaluating the integral.

Topic : Volume of a Solid

On Integration also one can find out volume of any solid.

Question : Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis x + y² = 16

Solution :
We know that the volume of the solid formed by rotating the area between the curve of f(y)
and the lines y = a and y = b about the x-axis is given by,




Given curve is x + y² = 16 Or x = 16 - y²
The graph of the line “x = 16 - y²” between the two axes is denoted by the shaded region in the graph drawn below:

















So, the volume of the shaded region when revolved around x-axis is given by:



















Hence, the volume of the solid generated by revolving the curve “x + y² = 16” about the x-axis is “128π”.

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