Answer
$$e - \frac{1}{e} + \frac{10}{3}$$
Work Step by Step
Area = $\int^d_c f(y) - g(y) dy$
In this question, $f(y) = e^y$ and $g(y) = y^2 - 2$
The limits of integration will equal the given lines $y = -1$ and $y = 1$
$$\int^d_c f(y) - g(y) dy$$
$$\int^1_{-1} (e^y)-(y^2 - 2) dy$$
Solve the produced definite integral and simplify until final answer.
$$= \int^1_{-1} (e^y - y^2 + 2)dy$$
$$= (e^y - \frac{y^3}{3} + 2y)|^1_{-1}$$
$$= [e^1 - \frac{1^3}{3} + 2(1)] - [e^{-1} - \frac{(-1)^3}{3} + 2(-1)]$$
$$= e - \frac{1}{3} + 2 - \frac{1}{e} - \frac{1}{3} + 2$$
$$= e - \frac{1}{e} + \frac{10}{3}$$
