# Volume of a solid w/known cross section

1. Oct 5, 2011

### 1MileCrash

1. The problem statement, all variables and given/known data

The base region of a solid is bounded by y=x, y=(x-1)^2, and x = 1.

The cross sections are semicircles perpendicular to the x-axis.

Write a riemann sum and definite integral.

2. Relevant equations

3. The attempt at a solution

First, I wrote down the formula for a semicircular disk's volume. 1/2(pi(r^2)(h))

I then found the intersection of y=x and y=(x-1)^2 to be .382 and another value that was greater than 1, so I ditched it.

I then wrote down the diameter of any given disk as x - (x-1)^2 or -x^2 + 3x - 3, so radius is half of that, and I defined the height of each disk to be delta x.

So, I wrote the Riemann Sum as: (limit as delta x approaches 0)

$\Sigma \frac{\pi}{4}(-x^{2}+3x-3)^{2}\Delta x$

And therefore wrote a definite integral as:

$\frac{\pi}{4} \int^{1}_{.382} (-x^{2}+3x-3)^{2} dx$

Did I do this right?

2. Oct 5, 2011

### SammyS

Staff Emeritus
It looks good. The true value for the lower limit of integration is, $\displaystyle \frac{3-\sqrt{5}}{2}\,.$