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1MileCrash
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Homework Statement
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.
Homework Equations
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)
[itex]\Sigma \frac{\pi}{4}(-x^{2}+3x-3)^{2}\Delta x[/itex]
And therefore wrote a definite integral as:
[itex]\frac{\pi}{4} \int^{1}_{.382} (-x^{2}+3x-3)^{2} dx[/itex]
Did I do this right?