Exam 14: Multiple Integrals

Find a parametric representation of the surface in terms of the parameters r and $\theta$ , where (r, $\theta$ , z) are the cylindrical coordinates of a point on the surface z = 12xy.

Find by first interchanging the order of integration.

Evaluate

Find the center of gravity of the lamina enclosed by x = 0, x = 4, y = 0, and y = 3 if its density is given by $\delta$ (x, y) = 5(x + y²).

Evaluate

Sketch R and express as an equivalent double integral with order of integration reversed.

Use cylindrical coordinates to find the mass of the solid bounded below by and above by if its density is given by . A) B) C) D) E)

Find a parametric representation of the surface in terms of the parameters r and $\theta$ , where (r, $\theta$ , z) are the cylindrical coordinates of a point on the surface .

Find the volume of the region given by lying above the xy-plane.

Use a triple integral to find the volume of the solid in the first octant enclosed by z = y, y² = x, and x = 1.

Use a triple integral to find the volume of the solid enclosed by z = 0, y = x² - x, y = x, and z = x + 1.

Find the area of the region enclosed by y = -x and y = x², for -4 $\le$ x $\le$ -1.

The centroid of a rectangular solid in the first octant with vertices (0, 0, 0), (0, 0, 4), and (4, 4, 4) is

Find the Jacobian, ; x = 3uv + w, y = u + 2v + 3w, z = u - v + 6w + 11.

Use a double integral in polar coordinates to find the volume of the solid enclosed by the paraboloid z = 36 - x² - y² and z = 0.

Evaluate by first sketching R then reversing the order of integration.

Use a CAS to solve the problem.

Find the Jacobian, ; x = 3u + 2, y = uv.

Find the Jacobian, ; x = 7e ^uv , y = uv⁶.

Find the volume of the solid in the first octant bounded above by below by z = 0, and laterally by the circular cylinder .