Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y = 10x − x2, y = x

Answer 1

First please draw the graphs of the functions.

You will find that the region in question is between a 45° line and an upside-down parabola.

In general, we have a choice whether to integrate over the x-axis, or the y-axis.

The suggestion to use cylindrical shells means that we should integrate over the

x-axis, adding up infinitesimally thin cylindrical shells of thickness dx.

A shell "at a position x" is the result of rotating the vertical line segment between

y1 = 10x-x² and y2 = x

around the y-axis, the rotation having radius x.

Therefore the volume of the shell at position x has the volume

2πx(y1-y2)dx =

2πx(10x-x²-x)dx =

2π(9x²-x³)dx

Adding up all these shells involves calculating the indefinite integral

V(x) = ∫2π(9x²-x³)dx =

2π(∫9x²dx - ∫x³dx) =

2π(3x³ - x⁴/4) + C =

2πx³(3 - x/4) + C

The next step is to calculate the bounds of integration.

In this case they are the intersections between the two curves:

10x - x² = x

There are two solutions: x = 0, and x = 9

Those are the bounds of integration.

Therefore the sought volume is

V(9) - V(0) = 2π9³(3 - 9/4) ≈ 3435