Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Use the table of integrals to find the exact area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and approximate the area. $$ y=x^{2} \sqrt{x^{2}+4}, y=0, x=\sqrt{5} $$

Step-by-Step Solution

Answer

The integral for the exact area bounded by \(y = x^{2} \sqrt{x^{2}+4}\), \(y = 0\), and \(x = \sqrt{5}\) from \(x = 0\) to \(x = \sqrt{5}\) is \(\int_{0}^{\sqrt{5}} x^{2} \sqrt{x^{2}+4} dx\) which evaluates to \(\frac{2}{3}[(9)^{3/2} - (4)^{3/2}]\). For getting the approximate area using a graph, utilize any graphing utility and note down the approximate area provided by it.

Step 1: Setting up the integral

The area bounded by the given curves is given by the definite integral of the equation \(y = x^{2} \sqrt{x^{2}+4}\) from \(x=0\) to \(x = \sqrt{5}\). Therefore the setup for the area bounded by the curves is \(\int_{0}^{\sqrt{5}} x^{2} \sqrt{x^{2}+4} dx\).

Step 2: Evaluating the integral

Evaluate the integral using appropriate techniques. For this integral, a good step could be to use the substitution method. Let \(u = x^{2}+4\), so that \(du = 2x dx\), and \(x dx = du/2\). The integral then becomes \(\frac{1}{2} \int u^{1/2} du\) from \(u=4\) to \(u = 9\). Evaluate this to get \(\frac{2}{3}[u^{3/2}]_{4}^{9}\).

Step 3: Computing the exact area

Substitute the value of the limits in the expression to find the value of the definite integral which represents the exact area under the curve. The exact area is thus \(\frac{2}{3}[(9)^{3/2} - (4)^{3/2}]\).

Step 4: Approximation using the graphing utility

In order to visualize the area under the curve, use a graphing utility to plot the function \(y = x^{2} \sqrt{x^{2}+4}\) and shade the area between the \(x-axis\), the function and \(x=\sqrt{5}\). Note down the approximate area provided by the utility.