Q. 24 Using polar coordinates to evalu... [FREE SOLUTION]

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Chapter 13: Q. 24 (page 1082)

Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.
${鈭�}_{- 5}^{0} {鈭�}_{- \sqrt{25 - x^{2}}}^{0} \frac{3}{{(4 + x^{2} + y^{2})}^{3}} d y d x$

Short Answer

Expert verified

${鈭�}_{- 5}^{0} {鈭�}_{- \sqrt{25 - x^{2}}}^{0} \frac{3}{{(4 + x^{2} + y^{2})}^{3}} d y d x = \frac{2475}{107648} 蟺$

Step by step solution

Draw the region

From the limits of integration, the region is shown below,

Convert into polar form

By using the following substitution,

$x = r \cos 胃 y = r \sin 胃 x^{2} + y^{2} = r^{2} d x d y = r d r d 胃$

The equivalent polar integral of the given integral is,

${鈭�}_{- 5}^{0} {鈭�}_{- \sqrt{25 - x^{2}}}^{0} \frac{3}{{(4 + x^{2} + y^{2})}^{3}} d y d x 鈬� {鈭�}_{蟺}^{\frac{3 蟺}{2}} {鈭�}_{0}^{- 5} \frac{3}{{(4 + r^{2})}^{3}} r d r d 胃$

Calculate the volume

"> $V = {鈭�}_{蟺}^{\frac{3 蟺}{2}} {鈭�}_{0}^{- 5} \frac{3}{{(4 + r^{2})}^{3}} r d r d 胃 V = {鈭�}_{蟺}^{\frac{3 蟺}{2}} d 胃 {鈭�}_{0}^{- 5} \frac{3}{{(4 + r^{2})}^{3}} r d r V = [\frac{3 蟺}{2} - 蟺] {鈭�}_{0}^{- 5} \frac{3}{{(4 + r^{2})}^{3}} r d r V = \frac{蟺}{2} {鈭�}_{0}^{- 5} \frac{3}{{(4 + r^{2})}^{3}} r d r Substitute, 4 + r^{2} = t 2 r d r = d t d r = \frac{1}{2 r} d t When r = 0, t = 4 When r = - 5, t = 4 + {(- 5)}^{2} = 29 V = \frac{蟺}{2} \frac{3}{2} {鈭�}_{4}^{29} \frac{1}{t^{3}} d t V = \frac{3 蟺}{4} {[\frac{t^{- 3 + 1}}{- 3 + 1}]}_{4}^{29} V = - \frac{3 蟺}{8} {[\frac{1}{t^{2}}]}_{4}^{29} V = - \frac{3 蟺}{8} [\frac{1}{841} - \frac{1}{16}] V = - \frac{3 蟺}{8} [\frac{16 - 841}{13456}] V = - \frac{3 蟺}{8} [- \frac{825}{13456}] V = \frac{2475}{107648} 蟺 cubic units$

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91影视

Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.
${鈭�}_{- 5}^{0} {鈭�}_{- \sqrt{25 - x^{2}}}^{0} \frac{3}{{(4 + x^{2} + y^{2})}^{3}} d y d x$

Short Answer

Step by step solution

Draw the region

Convert into polar form

Calculate the volume

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Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.鈭�-50鈭�-25-x2034+x2+y23dydx

Short Answer

Step by step solution

Draw the region

Convert into polar form

Calculate the volume

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Theoretical and Mathematical Physics

Calculus

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Using polar coordinates to evaluate iterated integrals: Evaluate the given iterated integrals by converting them to polar coordinates. Include a sketch of the region.
${鈭�}_{- 5}^{0} {鈭�}_{- \sqrt{25 - x^{2}}}^{0} \frac{3}{{(4 + x^{2} + y^{2})}^{3}} d y d x$