Question

find the area of the shaded region.

Explanation:

Step1: Set up integral for area

$A=\int_{0}^{1}(3e^{x}-3xe^{x^{2}})dx$

Step2: Integrate term - by - term

$\int_{0}^{1}3e^{x}dx=3e^{x}\big|_{0}^{1}=3e - 3$, $\int_{0}^{1}3xe^{x^{2}}dx=\frac{3}{2}e^{x^{2}}\big|_{0}^{1}=\frac{3}{2}(e - 1)$

Step3: Calculate the area

$A=(3e - 3)-\frac{3}{2}(e - 1)=\frac{3e}{2}-\frac{3}{2}$

Answer:

$\frac{3e}{2}-\frac{3}{2}$