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Chapter 13: Q 19. (page 1066)

Show that the mass of $鈭�$ is $\frac{1}{6} 蟺办$ by evaluating the integral:
$鈭� 鈭� {鈭�}_{鈭�} k d V = {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} k 蚁^{2} \sin 蠒 d 蚁 d 胃 d 蠒$

Short Answer

Expert verified

Use spherical coordinates $(蚁, 胃, 蠒)$ while evaluating $d V$ using triple integral.

Step by step solution

01

Given Information

Spherical coordinates are $(蚁, 胃, 蠒)$ . We need to prove this by solving given integral.

02

Simplification

Taking LHS.

$m = 鈭� 鈭� {鈭�}_{鈭�} k d V$

$= {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} k 蚁^{2} \sin 蠒 d 蚁 d 胃 d 蠒$

$= k {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{\frac{蟺}{2}} ({鈭�}_{0}^{1} 蚁^{2} \sin 蠒 d 蚁) d 胃 d 蠒$

$= k {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{\frac{蟺}{2}} {(\frac{蚁^{3}}{3})}^{1}_{0} \sin 蠒 d 胃 d 蠒$

$= k {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{\frac{蟺}{2}} \frac{1}{3} \sin 蠒 d 胃 d 蠒$

role="math" localid="1652246785701" $= \frac{k}{3} {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{\frac{蟺}{2}} \sin 蠒 d 胃 d 蠒$

role="math" localid="1652246883750" $= \frac{k}{3} {鈭�}_{0}^{\frac{蟺}{2}} {[胃]}^{\frac{蟺}{2}}_{0} \sin 蠒 d 蠒$

$= \frac{k}{3} {鈭�}_{0}^{\frac{蟺}{2}} [\frac{蟺}{2} - 0] \sin 蠒 d 蠒$

$= \frac{k 蟺}{6} {[- \cos 蠒]}^{\frac{蟺}{2}}_{0}$

$= - \frac{k 蟺}{6} [\cos \frac{蟺}{2} - \cos 0]$

$= - \frac{k 蟺}{6} [0 - 1]$

$= \frac{蟺办}{6}$

Hence, the mass is $\frac{1}{6} 蟺办$

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Most popular questions from this chapter

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.

$鈭� {鈭�}_{R} x e^{x y} d A, w h e r e {R = (x, y) | 0 鈮� x 鈮� 1 a n d 0 鈮� y 鈮� l n 5}$

Use Definition $13.4$ to evaluate the double integrals in Exercises $29 鈥� 32$ .

localid="1649936867482" ${鈭�}_{x} x^{2} y d A$

where $R = {(x, y) 鈭� 1 鈮� x 鈮� 3 and 0 鈮� y 鈮� 2}$

Let f(x, y, z) and g(x, y, z) be integrable functions on the rectangular solid $R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\}$ . . Use the definition of the triple integral to prove that :

${鈭�}_{R} (f (x, y, z) + g (x, y, z)) d V = {鈭�}_{R} f (x, y, z) d V + {鈭�}_{R} g (x, y, z) d V .$

$How many summands are in {鈭�}_{j = j_{0}}^{m} {鈭�}_{k = k_{0}}^{n} j^{k} ?$

Evaluate the sums in Exercises 23鈥�28.

${鈭�}_{j = 1}^{3} {鈭�}_{k = 1}^{2} k^{j}$

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