Answer to Question #155273 in Calculus for Izzah

Question #155273

Find the mass of the solid in the shape of the tetrahedron that is bounded by the planes x + y + z = 1 , xy- plane, xz- plane and yz- plane if its density is given by 𝛿(x, y, z) = x 


1
Expert's answer
2021-01-13T19:18:54-0500

The mass of the solid region "E" is evaluated as,


"m=\\iiint_{E}\\delta(x,y,z)dV" ,


where "E=\\lbrace(x,y,z)|0\\eqslantless x \\eqslantless 1, 0 \\eqslantless y \\eqslantless 1-x, 0 \\eqslantless z \\eqslantless1-x-y\\rbrace"


and "\\delta(x,y,z)=x"


So, the mass of the solid is,


"m=\\int_{0}^{1}\\int_{0}^{1-x}\\int_{0}^{1-x-y}(x)dzdydx"


"=\\int_{0}^{1}\\int_{0}^{1-x}x(1-x-y)dydx"


"=\\int_{0}^{1}x[(1-x)y-\\frac{y^2}{2}]_{y=0}^{1-x}dx"


"=\\int_{0}^{1}x[(1-x)(1-x)-\\frac{(1-x)^2}{2}]dx"


"=\\int_{0}^{1}x[(1-x)^2-\\frac{(1-x)^2}{2}]dx"


"=\\frac{1}{2}\\int_{0}^{1}x(1-x)^2dx"


"=\\frac{1}{2}\\int_{0}^{1}x(1+x^2-2x)dx"


"=\\frac{1}{2}\\int_{0}^{1}(x+x^3-2x^2)dx"


"=\\frac{1}{2}[\\frac{x^2}{2}+\\frac{x^4}{4}-2(\\frac{x^3}{3})]_{0}^{1}"


"=\\frac{1}{2}[\\frac{1}{2}+\\frac{1}{4}-\\frac{2}{3}]"


"=\\frac{1}{2}(\\frac{1}{12})"


"=\\frac{1}{24}"


Therefore, the mass of the solid region "E" is "m=\\iiint_{E}\\delta(x,y,z)dV=\\frac{1}{24}"

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

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS