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

Circular Motion and Gravitation

Further Mechanics and Thermal Physics

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Solid State Physics

Engineering Physics

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.
$\int_{x - 0}^{4} \int_{y - 0}^{x / 2} y - dy - dx$