Question

So I am not sure how to do this problem... As far as I know, you have to take the integral of cos^2(kx)*sin(kx). But apparently you have to take the derivative of sin(kx) and integrate cos^3(kx) as a result?

Answer 1

This is the work my friend sent me, but I cannot make sense of it at all.

Answer 2

You can start by writing the volume as a triple integral:
\(V = \displaystyle\iiint\limits_V \textrm{d}V = \int\limits_0^{\frac{\pi}{2k}}\int\limits_0^{\sin(kx)} \int\limits_0^{\cos^2(kx)}\textrm{d}z\textrm{ d}y\textrm{ d}x.\)
The integrals in \(y\) and \(z\) are trivial and indeed you get the expression your friend sent you:
\(V = \displaystyle\int\limits_0^{\frac{\pi}{2k}}\sin(kx)\cos^2(kx)\textrm{ d}x.\)
The simpler way is to integrate by substitution. Let \(u \equiv \cos(kx) \implies \textrm{d}u = -k\sin(kx)\textrm{ d}x\). Then the integration limits are from \(\cos (0k) = 1\) to \(\cos\left(\frac{\pi}{2k}k\right) = \cos\left(\frac{\pi}{2}\right) = 0\) and we can change the variable:
\(V = \displaystyle\int\limits_0^{\frac{\pi}{2k}}\underbrace{\cos^2(kx)}_{=u^2}\underbrace{\sin(kx)\textrm{ d}x}_{=-\frac{\textrm{d}u}{k}} = -\dfrac{1}{k}\int\limits_1^0 u^2\textrm{ d}u = \dfrac{1}{k}\int\limits_0^1 u^2\textrm{ d}u.\)
The integral is now really easy:
\(V = \dfrac{1}{k}\dfrac{u^3}{3}\Big\vert_0^1 = \dfrac{1}{k}\times\dfrac{1}{3} = \dfrac{1}{3k}.\)
Now we simply set \(V = 1\) to get:
\(V = 1 \iff \dfrac{1}{3k} = 1\iff k = \dfrac{1}{3} \qquad \boxed{\textrm{E}}.\)

Answer 3

I was thinking you could also use u-sub twice, perhaps it's a bit easier but harder to hold on to throughout the steps.

First let u=kx and then simplify. Then you can let v=cos(u), so you get -v^2 in the integrand and the sin(u) cancels. Then you can use the power rule in integration and plug cos(kx) back in. Then you have your antiderivative and thus you can evaluate.

Answer 4

It makes so much sense when u sub is introduced into this problem. I forget that that's a method that can be used since I barely use it myself. Thanks for the help guys!