Review question

# Can we find the area between $y=e^{-x}$ and $y=e^{-x}\sin x$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9312

## Suggestion

The curves $C_1$ and $C_2$ are defined by $y=e^{-x}\qquad (x>0)\qquad \text{and}\qquad y=e^{-x}\sin x \qquad (x>0),$ respectively. Sketch roughly $C_1$ and $C_2$ on the same diagram.

When does $e^{-x}\sin x$ equal $e^{-x}$?

Let $x_n$ denote the $x$-coordinate of the $n$th point of contact between the two curves, where $0<x_1< x_2<\dotsb$, and let $A_n$ denote the area of the region enclosed by the two curves between $x_n$ and $x_{n+1}$. Show that $A_n=\frac{1}{2}\left(e^{2\pi}-1\right)e^{-(4n+1)\pi/2}$

Where do the points $x_n$ occur?

If we look at the area that forms $A_1$ on your graph, how can we write that in terms of integrals? What about $A_n$?

How could we find $I = \int e^{-x}\sin x \,dx$? Could we use integration by parts twice?

…and hence find $\displaystyle\sum\limits_{n=1}^{\infty} A_n$.

Could we write out the first few terms of the infinite sum $\sum\limits_{n=1}^{\infty} A_n$?

What’s the expression for the infinite sum in this case? Can we spot a geometric series anywhere? What’s the sum to infinity for this?