Find the mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$ . $\rho ( x , y ) = 3$
A) $\frac { 3 } { 2 }$
B) $2$
C) 27
D) $\frac { 27 } { 2 }$
E) None of these

Question

Quizplus · Accepted Answer

First, we need to find the limits of integration for both x and y. Since D is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$, we can set up the following limits:
$$\int_{1}^{4}\int_{y^2}^{y+2}3dxdy$$
Now we can solve the double integral:
$$\int_{1}^{4}\int_{y^2}^{y+2}3dxdy = \int_{1}^4[3(x)|_{y^2}^{y+2}]dy$$
$$=\int_{1}^{4}(3(y+2)-3y^2)dy=\frac{27}{2}$$
The mass will be equal to the volume of the lamina, which in this case is the double integral of the density function over the region D. Therefore, the mass is $27/2$.

Find the Mass of the Lamina That Occupies the Region x=y2x = y ^ { 2 }x=y2

Find the Mass of the Lamina That Occupies the Region $x = y ^ { 2 }$