Answer to Question #100912 in Calculus for Yenco

Question #100912
1. Use a double integral to find the volume under the plane z=2x+y and over the rectangle R={(x,y) : 8 ≤ x ≤ , 3 ≤ y ≤ 4}.

2. Use a double integral to find the volume of the solid enclosed by the surface z= x^2 and the planes x=0, x=6, y=6, y=0, and z=0.

3. Use double integration to find the volume of the solid bounded by the cylinder x^2 + y^2 = 9 and the planes z=0 and z= 3-x.
1
Expert's answer
2020-01-05T17:14:57-0500

1.

Volume below the function "z=f(x,y)" and above the region R is given by "\\iint \\limits _R f(x,y)dA"

"V= \\iint \\limits _R f(x,y)dA=\\int\\limits _8 ^{10}\\int \\limits _3^4(2x+y)dy \\;dx=\\intop\\limits _8^{10} \\big[2xy+\\frac{1}{2}y^2 \\big]_3^4 dx="

"=\\int \\limits _8^{10} (8x+8)-(6x+\\frac{9}{2}) dx=\\int\\limits _8^{10} (2x+\\frac{7}{2})dx=\\big[x^2+ \\frac{7}{2}x\\big]_8^{10}="

"= (100+35)-(64+28)=43"

Answer: 43.

2.

Using the same formula for "z=x^2, R=\\{(x,y):0 \\leq x \\leq 6, 0 \\leq y \\leq 6\\}"

"V= \\iint \\limits _R x^2dA=\\int\\limits _0 ^{6}\\int \\limits _0^6 x^2dy \\;dx=\\intop\\limits _0^{6} \\big[x^2y\\big]_0^6 dx= \\intop\\limits _0^{6} 6x^2 dx= \\big[2x^3\\big]_0^6= 432"

Answer: 432.

3.

Using the same formula for "z=3-x, R=\\{(x,y):x^2+y^2=9\\}"

"V= \\iint \\limits _R(3-x)dA=\\int\\limits _{-3} ^{3}\\int \\limits _{-\\sqrt {9-y^2}}^{\\sqrt {9-y^2}}(3-x) \\;dx \\;dy=\\intop\\limits _{-3}^{3} \\big[3x-\\frac{1}{2}x^2\\big]_ {-\\sqrt {9-y^2}} ^ {\\sqrt {9-y^2}} dy="

"\\int \\limits _{-3}^{3}6 {\\sqrt {9-y^2}}dy= 6 * (\\pi 3^2)\/2=27\\pi"




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