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Chapter 13: Problem 20

Evaluate the double integral. $$ \int_{0}^{4} \int_{0}^{x} \frac{2}{x^{2}+1} d y d x $$

Short Answer

Expert verified
The value of the double integral is \( \ln 17 \).

Step by step solution

01

Simplify the inner integral

Since the integrand doesn't depend on y, you can perform the integration with respect to it directly. This procedure results in a simple multiplication of the integrated function by the difference of the limits of integration: \( (x - 0) \cdot \frac{2}{x^{2}+1} = \frac{2x}{x^{2}+1} \)
02

Compute the outer integral

Now, integrate the resulting function, \( \frac{2x}{x^{2}+1} \), with respect to x from 0 to 4. This integral is a basic one requiring substitution. If you let \( u = x^{2}+1 \), \( du = 2x dx \), the integral becomes \( \int u^{-1} du \), which integrates to \( \ln |u| \). Substituting back in for u gives \( \ln |x^{2} + 1| \). Evaluate this at the limits 0 to 4 to get the integral value.
03

Evaluate limits

Substitute the upper and lower limits to calculate the integral: \( \ln |4^{2} + 1| - \ln |0^{2} + 1| = \ln 17 - \ln 1 \). Since \( \ln 1 = 0 \), the answer simplifies to \( \ln 17 \).

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Most popular questions from this chapter

Exercises 55 and 56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{-1}^{1} \int_{-2}^{2} y d y d x=\int_{-1}^{1} \int_{-2}^{2} y d x d y $$

Sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{1}^{2} \int_{2}^{4} d x d y $$

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{1} \int_{x}^{1} \sqrt{1-x^{2}} d y d x $$

Sketch the region of integration and evaluate the double integral. $$ \int_{0}^{a} \int_{0}^{\sqrt{a^{2}-x^{2}}} d y d x $$

A firm's weekly profit in marketing two products is given by \(P=192 x_{1}+576 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-5000\) where \(x_{1}\) and \(x_{2}\) represent the numbers of units of each product sold weekly. Estimate the average weekly profit if \(x_{1}\) varies between 40 and 50 units and \(x_{2}\) varies between 45 and 50 units.
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