set up the integral to find the area of the shaded region in terms of (a) one or more integrals with respect…

Explanation:

Step1: Find area with respect to dx

The upper - curve is $y = e^{x}$ and the lower - curve is $y=x^{5}$ from $x = 0$ to $x = 1$. The formula for the area between two curves $y = f(x)$ and $y = g(x)$ is $A=\int_{a}^{b}(f(x)-g(x))dx$. So, $A_{dx}=\int_{0}^{1}(e^{x}-x^{5})dx$.

Step2: Find area with respect to dy

First, we need to express $x$ in terms of $y$. For $y = e^{x}$, we have $x=\ln y$, and for $y = x^{5}$, we have $x = y^{\frac{1}{5}}$. The area between two curves $x = h(y)$ and $x = k(y)$ is $A=\int_{c}^{d}(h(y)-k(y))dy$. The limits of integration for $y$ are from $y = 0$ to $y = e$. So, $A_{dy}=\int_{0}^{1}(y^{\frac{1}{5}}-0)dy+\int_{1}^{e}(\ln y - 0)dy$.

Answer:

(a) $\int_{0}^{1}(e^{x}-x^{5})dx$ (b) $\int_{0}^{1}y^{\frac{1}{5}}dy+\int_{1}^{e}\ln ydy$