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Chapter 13: Problem 14

Evaluate the double integral. $$ \int_{0}^{1} \int_{0}^{x} \sqrt{1-x^{2}} d y d x $$

Short Answer

Expert verified
The value of the double integral is -1.

Step by step solution

01

Understand the Iterated Integral

An iterated integral like \(\int_{0}^{1} \int_{0}^{x} \sqrt{1-x^{2}} dy dx\) involves performing multiple integrations, one after the other. The inner integral is done first, with respect to \(y\), then the result is integrated with respect to \(x\). The limits of the integral for \(y\) are from 0 to \(x\), and for \(x\) are from 0 to 1.
02

Perform the Inner Integral

The inner integral is \(\int_{0}^{x} \sqrt{1-x^{2}} dy\). The limits of integration are \(y = 0\) to \(y = x\). The function to integrate does not contain \(y\), so the integral simplifies to \([y\sqrt{1-x^{2}}]_{0}^{x}\), equal to \(x\sqrt{1-x^{2}} - 0\), which further simplifies to \(x\sqrt{1-x^{2}}\).
03

Perform the Outer Integral

We now need to perform the outer integral, which is \(\int_{0}^{1} x\sqrt{1-x^{2}} dx\). Here, the suggestion is to use the substitution method. Let \(u = x^{2}\) (thus \(du = 2xdx\)), then the integral becomes \(-\frac{1}{2} \int_{0}^{1} \sqrt{1-u} du\), where \(\sqrt{1-u}\) is the integral of \(u^{-1/2}\). The antiderivative of \(u^{-1/2}\) is \(2u^{1/2}\), so the evaluation of the definite integral becomes \(-\frac{1}{2} [2u^{1/2}]_{0}^{1}\), which simplifies to \(-1 + 0\).

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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. $$ \int_{2}^{5} \int_{1}^{6} x d y d x=\int_{1}^{6} \int_{2}^{5} x d x d y $$

The Cobb-Douglas production function for an automobile manufacturer is \(f(x, y)=100 x^{0.6} y^{0.4}\) where \(x\) is the number of units of labor and \(y\) is the number of units of capital. Estimate the average production level if the number of units of labor \(x\) varies between 200 and 250 and the number of units of capital \(y\) varies between 300 and 325 .

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.

Use a double integral to find the area of the region bounded by the graphs of the equations. $$ y=x, y=2 x, x=2 $$

Use a double integral to find the volume of the solid bounded by the graphs of the equations. $$ z=x+v, x^{2}+v^{2}=4 \text { (first octant) } $$
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