Question

find the area of the shaded region.
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Explanation:

Step1: Set up integral for area

$A=\int_{0}^{1}(7e^{x}-7xe^{x^{2}})dx$

Step2: Integrate term - by - term

$\int_{0}^{1}7e^{x}dx-\int_{0}^{1}7xe^{x^{2}}dx$
For $\int_{0}^{1}7e^{x}dx = 7e^{x}\big|_{0}^{1}=7e - 7$. For $\int_{0}^{1}7xe^{x^{2}}dx$, let $u = x^{2}$, $du=2xdx$, then $\int_{0}^{1}7xe^{x^{2}}dx=\frac{7}{2}\int_{0}^{1}e^{u}du=\frac{7}{2}(e - 1)$.

Step3: Calculate the area

$A=(7e - 7)-\frac{7}{2}(e - 1)=\frac{7e}{2}-\frac{7}{2}$

Answer:

$\frac{7e}{2}-\frac{7}{2}$