Solve the problem.
-Find the mass of a thin plate bounded by $| x | + | y | = 8$ if $\delta ( x , y ) = x ^ { 2 } y ^ { 2 }$.
A) $\frac { 32,768 } { 5 }$
B) $\frac { 262,144 } { 45 }$
C) $\frac { 262,144 } { 25 }$
D) $\frac { 32,768 } { 7 }$

Question

Quizplus · Accepted Answer

The given equation $|x| + |y| = 8$ represents a square in the xy-plane with vertices at (8,0), (0,8), (-8,0), and (0,-8). The density function is $\delta(x, y) = x^2y^2$. To find the mass of the plate, we integrate the density function over the area of the square. Due to the symmetry of the square and the density function, we can integrate over one of the four symmetrical regions and then multiply by 4. Let's choose the region where both $x$ and $y$ are positive, so $x + y = 8$ with $0 \leq x \leq 8$ and $0 \leq y \leq 8$.The mass $m$ is given by the integral:$$m = 4 \int_{0}^{8} \int_{0}^{8-x} x^2y^2 \,dy\,dx$$Solving the inner integral with respect to $y$:$$\int_{0}^{8-x} x^2y^2 \,dy = \frac{x^2y^3}{3}\Big|_{0}^{8-x} = \frac{x^2(8-x)^3}{3}$$Then, integrate with respect to $x$:$$m = 4 \int_{0}^{8} \frac{x^2(8-x)^3}{3} \,dx$$This integral, when solved, gives the mass of the plate. The correct solution to this integral is $\frac{262,144}{45}$, which corresponds to choice B. This involves standard techniques of polynomial integration and applying the limits of integration.

Solve the Problem ∣x∣+∣y∣=8| x | + | y | = 8∣x∣+∣y∣=8

Solve the Problem $| x | + | y | = 8$