[SOLVED] Volume of a Solid Revolution

yummycalc · June 13th 2006, 05:21 PM

I am having trouble with volumes using the definite integral. I wanted to place a couple of questions from my textbook. I was wondering does anyone have any good tips. I have never been good at visualizing volumes even if I sketch the graph on the paper. If anyone could help me I would really appreciate it. Here are two examples from my textbook:

Region bounded by square root of x, y= 2 and x= 0 about (a) the y-axis; (b) x=1

y= e^x, x = 0, x = 2 and y = 0 about (a) the y-axis; (b) y = -2. Estimate numerically.

Thanks very much.

**ThePerfectHacker** · June 13th 2006, 05:33 PM

Originally Posted by yummycalc

Region bounded by square root of x, y= 2 and x= 0 about (a) the y-axis; (b) x=1

Okay you have the following graph handdrawn below.

The point where the graphs intersect is at,
$x=4$ .
You can look at the graph or solve,
$\sqrt{x}=2$ --->by squaring both sides.

Thus, you are going to subtract from area of top curve the area of bottom curve.
$\mbox{Top curve: }y=2$
$\mbox{Bottom curve: }y=\sqrt{x}$

Now you take the are from $0\leq x\leq 4$
Finally by the volume of revolution you need to square and multiply by $\pi$ thus,
The integral is,
$\pi \int_0^4 \left( 2-\sqrt{x})^2 dx$

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