Chapter 7.3, Problem 34E

To determine

To find the area enclosed between the regions $y=x^2$ and $y=\left(x^2+x+1\right)e^{-x}$

Answer to Problem 34E

The area between the regions is $0$

Explanation of Solution

Given information: $y=x^2$ and $y=\left(x^2+x+1\right)e^{-x}$

Formula used: ${\displaystyle \int uvdx=u{\displaystyle \int vdx-{\displaystyle \int \left(u^{\prime }{\displaystyle \int vdx}\right)}}}dx$

Calculation :

The curve $y=x^2$ is a polynomial and the curve $y=\left(x^2+x+1\right)e^{-x}$ is of exponential is nature. For finding the area between the regions, there must be bounded region among the regions.

The given curves have no common region between the point of intersections.

Hence, area between the curves is $0$