<p>I'm entering the Ap exam Calculus BC and I am having a difficulty in solving some problems. I would appreciate the help.</p>
<p>1- Give the Volume of the solid generated by revolving the region bonded by the graph of y= In(x), the x-axis, the lines x = 1 and x = e, about the y axis.</p>
<p>Using the shell method you would form the integral from 1 to e of (2pi)xln x dx</p>
<p>This integral can be done easily by integration by parts.</p>
<p>Remarks:</p>
<p>(1) Always draw a picture first
(2) Decide if disk or shell will be easier - if you’re not sure just try one, and see if it works - you can always change your mind.
(3) In the case of the shell method, make sure you understand what the radius and height of a typical cylinder looks like.
(4) Write out the integral using the fomula for the area of a cylinder (2pi) rh
(5) Evaluate the integral
(6) The integral of xln x comes up enough that it is worth memorizing: it’s xln x - x (plus an arbitary constant, of course).</p>
<p>If you need further help understanding the problem, or computing the integral, let me know.</p>
<p>Thank you so much, I found this very helpful.
I had a problem in computing the Integral. I used integration by parts, from 1 to e of (2pi)xln x dx , but didn’t get the exact answer as from the multiple choice question.</p>
<p>Let u = ln x, dv = x. Then du = dx/x, and v = x^2/2. So uv - int v du = (x^2 ln x)/2 - 1/2 int x dx = (x^2 ln x)/2 - x^2/4.</p>
<p>By the way I have a typo in (6) above. It should say:</p>
<p>(6) The integral of ln x comes up enough that it is worth memorizing: it’s xln x - x (plus an arbitary constant, of course).</p>
<p>Knowing that allows you to do the IBP another way - by letting u=x, and dv = ln x dx. It’s a little quicker, but you need to be a bit more clever, since you have to solve an equation at the end.</p>
<p>Thanks a lot for the help. </p>
<p>In this one, I don’t know which one to u substitute with : integral of e^(2x) (tan(e^(2x)))^2 dx.</p>
<p>Thanks in advance.</p>
<p>u=e^(2x).</p>
<p>This is a very straightforward substitution. The derivative of e^(2x) is 2e^(2x) which appears in the integral (except the constant, which can always easily be taken care of).</p>