I have huge difficulties with integration (I'm decent with derivatives, thank god), and finding volumes and other applications of integration confuses the heck out of me. The prof really doesn't explain why things are the way they are, which I tend to need to fully understand how to do the question as well as questions like it.
The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross sections perpendicular to the x axis between these planes are squares whose bases run from the semi circle y= -sqrt(1-x^2) and y= sqrt(1-x^2). Find the volume.
I know that the functions given create the top and lower half of a circle, respectively. What I am really confused about with this question is how to draw the solid it asks for. I am told that the cross sections create squares... What do I do with this information?
The region enclosed by x= y^(3/2), x=0, y=2, rotated about the y axis.
So, I solve for the equation in terms of y, and I get y= x^(2/3) (first off, is this legal to do?). When I rotate this around the y axis, I get a cone-like shape. I know that the formula for finding the volume is the integral of pi times the radius squared, evaluated from y=0 to y=2, except that the radius is in terms of x. How should I go about solving this question?
A third one is:
Find the volume of the solid generate by revolving the regions bounded by the lines and curves about the x axis.
y=secx, y=tanx, x=0, x=1
I can't even picture this when it is made...
I can generally do the integration itself, I just have great difficulties developing the problem (including sketching certain functions, and how the data given to me fits into certain formulas).
Can anyone please help me work my way through this stuff? I really need to learn how to do it, and its just confusing the heck out of me... Thanks.