Question

Let R be the region between the graph of y=e^-2x and the x-axis for x is greater than or equal to 3. The are of R is

Answer 1

Do you know how to set up this problem?

Answer 2

We can express this as an improper integral \(\int_0^\infty e^{-2x}\,\mathrm{d}x\).$$\begin{align*}\int_0^\infty e^{-2x}\,\mathrm{d}x&=\lim_{a\to\infty}\int_0^a e^{-2x}\,\mathrm{d}x\\&=\lim_{a\to\infty}\left[-\frac12e^{-2x}\right]_0^a\\&=\frac12\lim_{a\to\infty}\left[e^{-2x}\right]_a^0\\&=\frac12\lim_{a\to\infty}\left(1-e^{-2a}\right)\\&=\frac12\end{align*}$$