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Chapter 13: Q 46. (page 1039)

$C = \{(x, y) 鈭� x^{2} + y^{2} 鈮� 1\}$
If the density at each point in $C$ is proportional to the point鈥檚 distance from the $y$ -axis, find the mass of $C$ .

Short Answer

Expert verified

The mass of region is $m = \frac{4}{3} k$

Step by step solution

01

Given Information

We are given that $C = \{(x, y) 鈭� x^{2} + y^{2} 鈮� 1\}$

02

Calculating Mass of Region

Mass of region is given by $m = {鈭�}_{惟} 蚁 (x, y) d A$

$蚁 (x, y) = k x$

$m = {鈭�}_{- 1}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k x d y d x$

According to figure

$m = 4 脳 mass of a quadrant$

$m = 4 {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - x^{2}}} k x d y d x$

$m = 4 {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - x^{2}}} k x d y d x$

Calculating inner integral first

$m = 4 {鈭�}_{01}^{1} k x [y]_{0}^{\sqrt{1 - x^{2}}} d x$

$m = 4 k {鈭�}_{0}^{1} x \sqrt{1 - x^{2}} d x$

Put $1 - x^{2} = t^{2}$

$鈬� - 2 x d x = 2 t d t$

$m = 4 k {鈭�}_{0}^{1} t^{2} d t = 4 k {[\frac{t^{3}}{3}]}_{0}^{1} = 4 k [\frac{1}{3}]$

Mass of region is $m = \frac{4}{3} k$

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Most popular questions from this chapter

Explain how the Fundamental Theorem of Calculus is used in evaluating the iterated integral ${鈭�}_{c}^{d} {鈭�}_{a}^{b} f (x, y) d x d y$ .

$L e t R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\} . P r o v e t h a t : {鈭�}_{R} d V = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) . W h a t i s t h e r e l a t i o n s h i p b e t w e e n R a n d t h e p r o d u c t (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) .$

In the following lamina, all angles are right angles and the density is constant:

Evaluate the iterated integral :

${鈭�}_{0}^{1} 鈥� {鈭�}_{0}^{3} 鈥� {鈭�}_{0}^{\sin^{鈭� 1} 鈦� x} 鈥� (x + y) d z d y d x$

Evaluate the sums in Exercises $23 鈥� 28$ .

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

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