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Chapter 12: Problem 37

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

Expert verified
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

01

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: \( \int_0^8 \frac{x}{\sqrt{x+1}} dx \) .
02

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: \( \int_1^9 (u^{1/2} - u^{-1/2}) du \).
03

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 \[\frac{2}{3}u^{3/2} - 2u^{1/2} |_1^9\] .
04

Substituting Upper and Lower Boundaries

We can now substitute the upper and lower boundaries. The exact area under the curve then will be: \[\frac{2}{3}* 9^{3/2} - 2*9^{1/2} - (\frac{2}{3}*1^{3/2} - 2*1^{1/2}) \]
05

Computing the Numerical Result

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

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Most popular questions from this chapter

Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{2}^{5} 10 x^{2} e^{-x} d x $$

Arc Length A fleeing hare leaves its burrow \((0,0)\) and moves due north (up the \(y\) -axis). At the same time, a pursuing lynx leaves from 1 yard east of the burrow \((1,0)\) and always moves toward the fleeing hare (see figure). If the lynx's speed is twice that of the hare's, the equation of the lynx's path is \(y=\frac{1}{3}\left(x^{3 / 2}-3 x^{1 / 2}+2\right)\) Find the distance traveled by the lynx by integrating over the interval \([0,1]\).

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{1}{x^{2}} d x $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{9} \frac{1}{\sqrt{9-x}} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} e^{-x^{2}} d x, n=2 $$
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