Question

Use a graphing utility to evaluate the integral. Graph the region whose area is given by the definite integral. $$ \int_{0}^{4} \frac{x}{\sqrt{2 x+1}} d x $$

Short answer

The definite integral of the given function from 0 to 4 is 2. This value represents the area under the curve of the function \(\frac{x}{\sqrt{2x+1}}\) from \(x=0\) to \(x=4\).

Step-by-step solution

Rearrange the integrand

Rewrite the integrand to \(\frac{x}{\sqrt{2x+1}}dx\) as \(\frac{1}{2} \cdot (2x+1)^{1/2-1}dx\). This is done to see if the integrand can be expressed in the form \(f’(x).f(x)^n\), which fits the formula for the integration by substitution method.

Apply integration by substitution

In integration by substitution, the integral of a function can be computed by replacing it with a simpler function. Here, let \(u=2x+1\), then differentiating \(u\) gives \(du = 2dx\) and \(dx=du/2\). Replacing \(x\) in the bounds with \(u\), when \(x=0\), \(u = 1\), and when \(x=4\), \(u = 9\). Substitute these expressions into the integral: \(\int_{1}^{9} \frac{1}{2}u^{1/2-1}du\).

Evaluate the integral

This integral can be solved using the power rule for integrals, it gives, \(\frac{1}{2} \cdot \frac{u^{1/2}}{1/2}\) evaluated from 1 to 9. The limits 1 and 9 are then substituted into the solved integral to calculate the definite integral, resulting in \([3-1]=2\).

Graph the function and observe the area

The function \(\frac{x}{\sqrt{2x+1}}\) is plotted using a graphing utility from \(x=0\) to \(x=4\). The area under the curve and above the x-axis between \(x=0\) and \(x=4\) corresponds to the definite integral and comes out to be equal to the result, 2.