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. $$ y=\frac{x}{\sqrt{x+1}}, y=0, x=8 $$

Short answer

The exact area under the curve \(y = \frac{x}{\sqrt{x+1}}\) from x = 0 to x = 8 is 12 square units.

Step-by-step solution

Expression of the Problem as a Definite Integral

Since the area under the curve of a non-negative, continuous function from a to b can be calculated as the definite integral of the function from a to b, we can express the problem as follows: \( \int_0^8 \frac{x}{\sqrt{x+1}} dx \) .

Computation of the Definite Integral

The computation of the integral can be simplified by using the substitution method. We set \(u = x + 1\). Then, \(du = dx\), \(x = u -1\), and we adjust the limits of the integral to match the new variable. The integral then becomes \( \int_1^9 \frac{u-1}{\sqrt{u}} du \). This integral can be computed as: \( \int_1^9 (u^{1/2} - u^{-1/2}) du \).

Solving the Integral

Now we can split the integral and solve step by step: The integral of \(u^{1/2}\) with respect to u is \(\frac{2}{3}u^{3/2}\) and the integral of \(u^{-1/2}\) with respect to u is \(2u^{1/2}\). The resulting calculation brings us to \[\frac{2}{3}u^{3/2} - 2u^{1/2} |_1^9\] .

Substituting Upper and Lower Boundaries

We can now substitute the upper and lower boundaries. The exact area under the curve then will be: \[\frac{2}{3}* 9^{3/2} - 2*9^{1/2} - (\frac{2}{3}*1^{3/2} - 2*1^{1/2}) \]

Computing the Numerical Result

Finally we can come up with the exact number for the area under the curve: \[18 - 6 - (\frac{2}{3} - 2) = 12\]