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 $y = x - 2$.
$$\rho ( x , y ) = 5$$
Select the correct answer.
A) $\frac { 45 } { 2 }$
B) $\frac { 5 } { 2 }$
C) 45
D) 3
E) None of these

Question

Quizplus · Accepted Answer

The mass of the lamina is found by integrating the density function over the region $D$. The region $D$ is bounded by the parabola $x = y^2$ and the line $y = x - 2$. Solving these equations simultaneously gives the points of intersection, which are $(4, 2)$ and $(1, -1)$. The mass is calculated by setting up the double integral of the density function $\rho(x, y) = 5$ over the region $D$, which results in $\int_{-1}^{2} \int_{y^2}^{y+2} 5 \, dx \, dy$. Evaluating this integral gives the mass as $\frac{45}{2}$.

Find the Mass of the Lamina That Occupies the Region DDD

Find the Mass of the Lamina That Occupies the Region $D$