Answer to Question #100913 in Calculus for Jeffrey Chen

Question #100913
1. Use a double integral to find the volume V of the solid that is common to cylinders x^2 + y^2 = 81 and x^2 + z^2 = 81. Find the exact number and no tolerance

2. Use a triple integral to find the volume of the solid in the first octant bounded by the coordinate planes and the plane 9x+18y+8z = 144.

3. Use a triple integral to find the volume of the solid bounded by the surface y=x^2 and the planes x+z=8 and z=0.Give the exact answer in the form of a fraction.
1
Expert's answer
2020-01-12T14:20:42-0500

1."Volume = 8\\int ^9_0\\int^{\\sqrt{81-x^2}}_0\\int^{\\sqrt{81-x^2}}_0dydzdx"

"Volume=8\\int^9_0(\\sqrt{81-x^2})^2 dx"

"Volume=8\\int^9_0(81-x^2 )dx"

"Volume=8[81x-x^3\/3]^9_0"

"Volume=8(81\u00d79-81\u00d73)"

"Volume=81\u00d76\u00d78=3888(Ans)"

2.

"Volume=\\int ^{16}_0\\int^{8-x\/2}_0(16-(9\/8)x-(9\/4)y)dydx"

"Volume=\\int ^{16}_0[(16y-(9\/8)yx-(9\/8)y^2)]^{8-x\/2}_0"

"Volume=\\int ^{16}_0[(16(8-x\/2)-(9\/8)(8x-x^2\/2) -(9\/8)(8-x\/2)^2)]dx"

"Volume=[-4( 8-x\/2)^2-(9\/8)(4x^2-x^3\/6)+(3\/16)(8-x\/2)^3]^{16}_0=224(ans)"

3."Volume= \\int^{64}_0\\int_ {\\sqrt{y}}^8\\int_0^{8-x}dzdxdy"

"Volume= \\int^{64}_0\\int_ {\\sqrt{y}}^8(8-x)dxdy"

"Volume= \\int^{64}_0[(8x-x^2\/2)]^8_{\u221ay}dy"

"Volume= \\int^{64}_0(32-(8\u221ay)+y\/2)dy"

"Volume= (32y-(8y^{3\/2})+y^2\/4)]^{64}_{0}"

"Volume= (32\u00d764-8^4+64^2\/4)]=1024(Ans)"






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