1.18: Past-paper question 9

Figure 3 shows a sketch of part of the curve $\displaystyle S$ with equation $\displaystyle y = -2\mathrm{e}^{3x} + 4$ and the line $\displaystyle L$

The curve $\displaystyle S$ has intersections with the line $\displaystyle L$ at the points $\displaystyle A$ and $\displaystyle B$ with $\displaystyle x$ coordinates $\displaystyle x = -1$ and $\displaystyle x = 0$ respectively.

The finite region bounded by $\displaystyle S$ and $\displaystyle L$ is shown shaded in Figure 3

Use calculus to find the exact area of this region.

Give your answer in the form $\displaystyle \frac{a + be^{-c}}{c}$ where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers to be found.