I need help with 2 questions on volumes. Any help would be greatly appreciated!!
1.) y=x^2 , y=1 , x=0 , about y=2
2.) y=x , y= square root of x , about x=1
First draw a picture showing $\displaystyle y = x^2$, $\displaystyle y = 1$, & $\displaystyle x = 0$ (I usually use a different color pen for each curve, makes it easier to identify them) Then draw a dotted line for y = 2 since this is the line you will be rotating around.
So you have a plain 'ole parabola, a line at x=0 (the y-axis) and a line at y=1. Shade in the area bounded by those curves. For this problem, it looks like you can probably use washers or cylindrical shells. I'd probably use washer method.
So your integral will look like:
$\displaystyle \int\limits^{1}_{0} \pi \cdot(2-x^2)^2 - (2-1)^2 dx$
Since you are rotating about another axis, you need to account for that. You subtract each radius from the new axis of rotation.
* Disks: $\displaystyle \int \pi \cdot ((outer radius)^2 - (inner radius)^2) dx$
Then, just integrate and evaluate at the limits. Can you try the second one on your own? Good luck!!