Question

set up the integral to find the area of the shaded region in terms of (a) one or more integrals with respect to dx. (b) one or more integrals with respect to dy. you need to enter your setup and answer for both part (a) and (b). click insert > equation

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) $\int_{0}^{1}y^{\frac{1}{5}}dy+\int_{1}^{e}\ln ydy$