Find the area bounded by the parametric curve $x = \cos(t)$, $y = e^t, 0 < t < \pi/2$, and the lines $y = 1$ and $x = 0$.
I do not even know where to start with this problem. I know that I need to draw a graph, but that's all I know. Thanks for the help!
Best Answer
If $x=\cos(t)$, then $t=\arccos(x)$, at least when $0\leq t \leq \pi/2$. Thus, your curve can be expressed as $y=e^{\arccos(x)}$, which you can integrate over $[0,1]$.
I certainly agree that a picture like the one below is useful. You could plot both the parametric plot and the function plot using a tool like WolframAlpha. It then becomes apparent that you should subtract 1 from the integral. Of course, you can also ask WolframAlpha for help with the integral, which might be a bit tricky.