Question

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.
x = 5 + y2, x = 0, y = 2, y = 3

Answer 1

The volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis will be V = 149.2.

From the question.

x = 5 + y2,

x = 0,

y = 2,

y = 3

What is the equation for the volume?

The equation for the volume is mathematically given as

$V = \int\limits^3_2 (2\pi y)(5+y^2) \, dy \\\\\\V = 2\pi \int\limits^3_2(5y+y^3) \, dy \\\\\\V = 2\pi \frac{5y^2}{2} + \frac{y^4}{4} |^3_2\\\\\\V = 149.2$

Therefore

The volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis will be V = 149.2.

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