Ex. 1. Use a triple integral to find the volume of the tetrahedron bounded by the four planes x = 0, z = 0, x = 2y, and x + 2y + z = 2.

(b) The order of integration can make the problem much easier or harder. (c) The triple integral of gives you the volume of the region of integration.

EXAMPLE 2 Find the volume of a tetrahedron (4-sided pyramid). Locate (2,j,5). Solution A tetrahedron has four flat faces, all triangles. The fourth face in.

Like dA = dxdy was a symbol for a small area, dV = dxdydz indicates a small volume. Triple integrals are solved as a nested list of single integrals.

Example6. Find the volume under the paraboloid within the cylinder. Example7. Evaluate the integral over the region which is bounded between and. Example8.

a) x = 0, x = 1, y = 0, y = 3, z = 0, z = 2 ρ(x, y, z) = x + y + z answer: 18 b∗) x2 + y2 = 4, z = 0, z = 2px2 + y2 ρ(x, y, z) = z answer: 16π.

Average value of a function in a region in space. Example. Find the average of f (x,y,z) = xyz in the first octant bounded by the planes x = 1, y = 2, ...

Let U be the solid contained in x2 + y2 − z2 = 16 and lying between the planes z = −3 and z = 3. Sketch U and write an iterated integral which expresses its ...

(2x – y) is integrated with respect to x between x = 2 and x = 3, with y regarded as a constant i.e. (x – 5y) is integrated with respect to x between x = 1 and ...

EXAMPLE 3 A box corresponding to [0; 1] [0; 1] [0; 2] in xyz- coordinates is filled with a mixture of small machine parts (heavy) and styrofoam peanuts (light) ...