Volume of solid revolving about y-axis

[frumdogg]

Homework Statement

Find the volume of the solid generated by revolving the region bounded by the graph of
y = x3 and the line y = x,
between x = 0 and x = 1,
about the y-axis.

Homework Equations

$$\pi$$$$\overline{1}$$$$\int$$$$\underline{0}$$[R(x)$$^{2}$$-[r(x)]$$^{2}$$dx

The Attempt at a Solution

x^6 - x^2 dx = x^7/7 - x^3/3 is where I get stuck.

 

[Alienjoey]

Since you're revolving it around the y axis, you would probably want to integrate the areas with respect to y. Once you do that and find your new limits of integration, there shouldn't be much of a problem getting the answer.

Edit: And don't forget about that Pi in the equation, when you integrated with respect to x you omitted it.

 

[tron_2.0]

yes you want each slice perpendicular to the line you are rotating to. in your case, each slice will be (deltaY) high so you would integrate in terms of y, not x.

then it just becomes the integral of pi(Routside)^2-pi(Rinside)^2dy