/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 23

Evaluate the following integrals as they are written. $$\int_{-2}^{2} \int_{x^{2}}^{8-x^{2}} x d y d x$$

Short Answer

Expert verified
In this problem, we have computed the value of the given double integral. The step-by-step solution was provided by first integrating with respect to y, followed by integrating with respect to x. The final result of the integral is 0.

Step by step solution

01

Integrate with respect to y

We are given the integral $$\int_{-2}^{2} \int_{x^{2}}^{8-x^{2}} x d y d x$$, so we start by integrating with respect to y: $$\int_{-2}^{2} [ \int_{x^{2}}^{8-x^{2}} x d y ] d x$$ The antiderivative of x with respect to y is xy: $$\int_{-2}^{2} [ xy\Big|_{x^2}^{8-x^2} ] d x$$
02

Find values of the antiderivative

Evaluate the antiderivative using the given bounds, and subtract in the order \(8-x^2\) (the higher bound) and then \(x^2\) (the lower bound): $$\int_{-2}^{2} [ (x(8-x^2)) - (x(x^2)) ] d x$$ Simplify the expression inside the integral: $$\int_{-2}^{2} [ 8x-x^3 - x^3] dx$$ Combine like terms: $$\int_{-2}^{2} [ 8x-2x^3 ] dx$$
03

Integrate with respect to x

Now we integrate the expression with respect to x: $$\int_{-2}^{2} [ 8x-2x^3 ] dx$$ The antiderivative of \(8x-2x^3\) with respect to x is \(4x^2-\frac{1}{2}x^4\): $$[4x^2 - \frac{1}{2}x^4 \Big|_{-2}^{2}]$$
04

Evaluate the integral

Finally, evaluate the integral using the given bounds and subtract the values to find the result: $$\Big(4(2)^2 - \frac{1}{2}(2)^4\Big) - \Big(4(-2)^2 - \frac{1}{2}(-2)^4\Big)$$ Calculate the values: $$(16-16)-(16-16)=0$$ So, the result of the given integral is 0: $$\int_{-2}^{2} \int_{x^{2}}^{8-x^{2}} x d y d x = 0$$

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