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Chapter 12: Problem 6

Find the mass and center of mass of the lamina for each density. \(R:\) triangle with vertices \((0,0),(0, a),(a, 0)\) (a) \(\rho=k\) (b) \(\rho=x^{2}+y^{2}\)

Short Answer

Expert verified
For density \(\rho = k\), the mass is \(M = \frac{1}{2}ka^2\) and the center of mass is \((\bar{x}, \bar{y}) = (\frac{a}{3}, \frac{a}{3})\). For density \(\rho = x^{2}+y^{2}\), the mass is \(M = \frac{1}{6}a^4\) and the center of mass is \((\bar{x}, \bar{y}) = (\frac{a}{4}, \frac{a}{4})\).

Step by step solution

01

find the mass with density \(\rho = k\)

The mass is equal to the integral over area R of density. We calculate the integral \(\int_0^a \int_0^{a-x} k dy dx\), where k is a constant. The mass (M) equals to \(M = \frac{1}{2}ka^2\).
02

find the center of mass with density \(\rho = k\)

We find the center of mass using the formula \((\bar{x}, \bar{y}) = (\frac{1}{M} \int_R x\rho(x,y) dA, \frac{1}{M} \int_R y\rho(x,y) dA)\). The center of mass equals to \((\bar{x}, \bar{y}) = (\frac{a}{3}, \frac{a}{3})\).
03

find the mass with density \(\rho = x^{2}+y^{2}\)

Now we calculate the mass using density function \(\rho = x^{2}+y^{2}\). We need to calculate the integral \(\int_0^a \int_0^{a-x} (x^{2}+y^{2}) dy dx\). This gives us \(M = \frac{1}{6}a^4\).
04

find the center of mass with density \(\rho = x^{2}+y^{2}\)

Using the formula for the center of mass like in step 2, we find that the center of mass equals to \((\bar{x}, \bar{y}) = (\frac{a}{4}, \frac{a}{4})\).

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Most popular questions from this chapter

In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{1} \int_{0}^{2}(x+y) d y d x $$

In Exercises \(43-50\), sketch the region \(R\) whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. $$ \int_{0}^{9} \int_{\sqrt{x}}^{3} d y d x $$

In Exercises \(11-22,\) evaluate the iterated integral. $$ \int_{0}^{1} \int_{0}^{\sqrt{1-y^{2}}}(x+y) d x d y $$

Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density or densities. (Hint: Some of the integrals are simpler in polar coordinates.) $$ y=9-x^{2}, y=0, \rho=k y^{2} $$

Find the mass of the lamina described by the inequalities, given that its density is \(\rho(x, y)=x y .\) (Hint: Some of the integrals are simpler in polar coordinates.) $$ x \geq 0,0 \leq y \leq \sqrt{4-x^{2}} $$
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