Question

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=x^{2}+2 x+1, y=3(x+1) $$

Short answer

The area enclosed between the two curves using double integral can be computed by following these steps.

Step-by-step solution

Identify Intersection Points

First, we identify the points where the two graphs intersect. We can do this by setting the two equations equal to each other then solve for \(x\). Thus, we get \(x^{2} + 2x + 1 = 3(x+1)\). Simplifying this to quadratic equation and solving, gives \(x=-1\) and \(x=1\).

Set up the Double Integral

Next, we set up the integral. We need to subtract the smaller function from the larger function within the double integral to determine the volume between the two curves. The bounds of \(x\) will be from \(-1\) to \(1\) (intersection points). Thus, the double integral can be set up as follows: \[ \int_{-1}^{1}\int_{3x+3}^{x^{2}+2x+1}dydx \]

Compute the Integral

We then perform the integration. Since we're integrating with respect to \(y\) first, this simply amounts to evaluating the \(y\) values at the respective \(x\), which will give us a simple \(x\) integral that we can then calculate. The evaluated integral will be the area enclosed between the two curves.