Don't really know how to solve

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

 y=3x, y=3√x;    about y=3 

Answer 1

Intersection Points

The curves y = 3x and y = 3√x intersect at x = 0 and x = 1

Intersection points are (0,0) and (1, 3)

Cross-Sectional Area Function

Volume is calculated by integration of a cross-sectional area function, A(x), over the interval between intersection points. Rotation about the x-axis (simpler) gives the same volume as rotating about y = 3, since they are both horizontal lines that pass through one of the two intersection points. The cross-sectional area for rotation about the x-axis is given by

A(x) = π[(3√x)² – (3x)²] = 9π[x –x²]

Volume

Volume is given by the integral

Answer 2

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