Question

Finding Area Find the area of the region enclosed by the \(y\) -axis and the curves \(y=x^{2}\) and \(y=\left(x^{2}+x+1\right) e^{-x}\)

Short answer

The steps mentioned provide a guide for solving the problem. Actual solution would require numerical integration as the points of intersection of the curves and the integrals may not have standard algebraic solutions.

Step-by-step solution

Find the Intersection Points

Set the two equations equal to each other: \(x^{2}=\left(x^{2}+x+1\right) e^{-x}\). Solve this equation for \(x\). These solutions represent the points where the two curves intersect.

Setup the Integral

The area of the region between the two curves from \(x=a\) to \(x=b\) is equal to \(\int_{a}^{b\}| f(x) - g(x) | dx\), where \(f(x)\) and \(g(x)\) are the two functions, and \(a\) and \(b\) are the intersection points found in Step 1.

Evaluate the Integral

Compute the integral \(\int_{a}^{b\}| f(x) - g(x) | dx\). Be mindful of the absolute value, which means that the integrand will need to be split into separate intervals where one function is greater than the other.

Adding the Areas

After computing the areas under the curves for all intervals, add them together to get the total area between the curves.