Triple Integral over a Tetrahedron

Example

Find the volume of the first-octant tetrahedron bounded by

x+y+z=1.x+y+z=1.

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Cylindrical Coordinates: Paraboloid Volume

Example

Find the volume bounded above by

z=4−x2−y2z=4-x^2-y^2

and below by z=0z=0.

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Spherical Coordinates: Radial Integrand

Example

Evaluate

∭Be(x2+y2+z2)3/2 dV\iiint_B e^{(x^2+y^2+z^2)^{3/2}}\,dV

over the unit ball BB.

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Center of Mass of a Solid Hemisphere

Example

A solid hemisphere x2+y2+z2≤R2x^2+y^2+z^2\le R^2, z≥0z\ge0 has constant density δ\delta. Find zˉ\bar z.

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Moment of Inertia of a Solid Cylinder

Example

A solid cylinder x2+y2≤4x^2+y^2\le4, 0≤z≤30\le z\le3 has constant density δ=1\delta=1. Find IzI_z.

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Complete Non-Polar Change of Variables

Example

Let RR be the parallelogram bounded by

x+y=0,x+y=2,x−y=0,x−y=1.x+y=0,\qquad x+y=2,\qquad x-y=0,\qquad x-y=1.

Evaluate

∬R(x2−y2) dA\iint_R (x^2-y^2)\,dA

using

u=x+y,v=x−y.u=x+y,\qquad v=x-y.

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Three-Dimensional Jacobian Example

Example

Use the transformation

x=2u,y=3v,z=4wx=2u,\qquad y=3v,\qquad z=4w

to find the volume of the rectangular box 0≤x≤20\le x\le2, 0≤y≤60\le y\le6, 0≤z≤40\le z\le4.

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