Chapter 5: Q4P (page 247)

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer.
$\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx$

Short Answer

Expert verified

Value of given integral is $\frac{8}{3}$ .

Step by step solution

Integrating the integral with respect to y first

Now using the by-parts method of integration for the given integral.

$\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - d y - d x = \int_{x - 0}^{4} (\int_{y - 0}^{x / 2} y - d y) d x$

Using the n-th rule of integralrole="math" localid="1658833631248" $(\int x^{n} - dx = \frac{x^{n + 1}}{n + 1} + C)$ .

role="math" localid="1658833730674" $\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx = \int_{x - 0}^{4} {(\frac{1}{2} y^{2})}_{0}^{x / 2} - dx$

Now put the upper and lower limit into the integral and calculate.

role="math" localid="1658833596213" $\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - d y - d x = \int_{x - 0}^{4} \frac{1}{2} ({(\frac{x}{2})}^{2} - 0^{2}) - d x \int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - d y - d x = \int_{x - 0}^{4} \frac{x^{2}}{8} d x$

Now integrating the integral get in step 1 with respect to x

Integrating with respect to x,

$\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - d y - d x = \int_{x - 0}^{4} \frac{x^{2}}{8} d x$

Now again using the n-th rule of integral $(\int x^{n} - dx = \frac{x^{n + 1}}{n + 1} + C)$ .

$\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx = {(\frac{x^{3}}{8 \times 3})}_{4}^{0} \int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx = (\frac{1}{24}) (4^{2} - 0^{3}) \int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx = \frac{64}{24} \int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx = \frac{8}{3}$

Therefore, the value of the given integral is $\frac{8}{3}$ .

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In the problems of this section, set up and evaluate the integrals by hand and check your results by computer.
$\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx$

Short Answer

Step by step solution

Integrating the integral with respect to y first

Now integrating the integral get in step 1 with respect to x

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In the problems of this section, set up and evaluate the integrals by hand and check your results by computer.∫x-04∫y-0x/2y-dy-dx

Short Answer

Step by step solution

Integrating the integral with respect to y first

Now integrating the integral get in step 1 with respect to x

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

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Study anywhere. Anytime. Across all devices.

In the problems of this section, set up and evaluate the integrals by hand and check your results by computer.
$\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx$