Solve the problem.
-Find the mass of a thin infinite region in the first quadrant bounded by the coordinate axes and the curve $y = e ^ { - 8 x }$ if $\delta ( x , y ) = x y$.
A) $\frac { 1 } { 512 }$
B) $\frac { 1 } { 192 }$
C) $\frac { 1 } { 256 }$
D) $\frac { 1 } { 96 }$

Question

Quizplus · Accepted Answer

The mass of the region can be found by integrating the density function $\delta(x, y) = xy$ over the given region. The integral for the mass $m$ is given by:$$m = \int_{0}^{\infty} \int_{0}^{e^{-8x}} xy \, dy \, dx.$$First, integrate with respect to $y$:$$m = \int_{0}^{\infty} \left( \frac{1}{2}xy^2 \right)_{0}^{e^{-8x}} dx = \int_{0}^{\infty} \frac{1}{2}x(e^{-8x})^2 dx = \int_{0}^{\infty} \frac{1}{2}x e^{-16x} dx.$$This is a standard integral that can be solved by parts or by recognizing it as a gamma function. The result of the integration is:$$m = \frac{1}{2} \cdot \frac{1}{(16)^2} = \frac{1}{512}.$$Thus, the mass of the region is $\frac{1}{512}$.

Solve the Problem y=e−8xy = e ^ { - 8 x }y=e−8x

Solve the Problem $y = e ^ { - 8 x }$