Answer to Question #343961 in Calculus for Nina

Question #343961

Find the mass of the lamina in the shape of the portion of the plane with equation 4x + 8y + z = 8 in the first octant if the area density at any point (x, y, z) on the plane is δ(x, y, z) = 6x + 12y + z g/cm^2


1
Expert's answer
2022-05-24T09:56:53-0400

ANSWER The mass of the lamina is 96g96g .

EXPLANATION

The mass of the part of the plane SS is calculated using surface integral according to the formula m=Sδ(x,y,z)dSm=\iint_{S}\delta(x,y,z)dS , where S={(x,y,z):z=84x8y,0x2,0y1x2}S= \left \{ \left ( x,y,z \right ): z=8-4x-8y,0\leq x\leq 2,\, 0\leq y\leq 1-\frac{x}{2} \right \} . SS has a one-to-one projection onto the domain DD in the xyxy- plane:

D={(x,y):0x2,0y1x2}D=\left \{ \left ( x,y \right ): 0\leq x\leq 2,\, 0\leq y\leq 1-\frac{x}{2}\right \} . The surface area element on SS is given by dS=1+zx2+zy2dxdy=1+16+64dxdy=9dxdydS=\sqrt{1+z_{x}^{2}+z_{y}^{2}}\, dxdy=\sqrt{1+16+64}dxdy=9 dxdy . The surface integral of δ(x,y,z)\delta (x,y,z) over SS can be expressed as a double integral over the domain DD

aminaSδ(x,y,z)dS=Dδ(x,y,84x8y)9dxdy==D(6x+12y+84x8y)9dxdy=18D(x+2y+4)dxdy==1802(01x2(x+2y+4)dy)dx=1802{[(x+2y+4)24]y=01x2}dx==1802(9(x+4)24)dx=18{18[(x+4)312]02}=18(186312+4312)=18163=96amina \iint_{S} \delta (x,y,z)dS=\iint_{D} \delta (x,y,8-4x-8y)\cdot 9dxdy=\\=\iint_{D} (6x+12y+8-4x-8y)\cdot 9dxdy=18\iint_{D} ( x+2y+4 ) dxdy= \\=18\int_{0}^{2}\left ( \int_{0}^{1-\frac{x}{2}}\left ( x+2y+4 \right )dy \right )dx=18 \int_{0}^{2}\left \{\left [ \frac{\left ( x+2y+4 \right )^{2}}{4} \right ]_{y=0}^{1-\frac{x}{2}} \right \}dx=\\=18 \int_{0}^{2}\left ( 9-\frac{(x+4)^{2}}{4} \right )dx= 18 \left \{ 18-\left [ \frac{(x+4)^{3}}{12} \right ]_{0}^{2} \right \}= 18(18- \frac{6^{3}}{12}+\frac{4^{3}}{12})=18\cdot \frac{16}{3}=96 .

So, m=96m=96


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment