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 y-axis
y = 2x, y = 4, x = 0

Solution :

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







The graph of the line “ y = 2x” and “y = 4” 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 y-axis is given by:





















Hence, the volume of the solid generated by revolving the curves “ y = 2x” and “y = 4” around y axis in the interval [0,2] is “16π /3”.


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