Exam 14: Multiple Integrals

Use a double integral in polar coordinates to find the volume in the first octant of the solid enclosed by x² + y² = 16, y = z, and z = 0.

Use a double integral in polar coordinates to find the area enclosed by the limacon r = 8 + sin $\theta$ .

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

Find the volume of the solid enclosed by x = 0, z = 0, z = 4 - x², y = 3x, and y = 9.

Evaluate

Evaluate . Hint: First convert to cylindrical coordinates.

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 and x = r cos $\theta$ , y = r sin $\theta$ on the surface is

Find the Jacobian if and .

Find the Jacobian, ; x = 5 + e ^uv , y = uv².

The surface expressed parametrically by , , and is:

Find the centroid of the lamina enclosed by y = 2(4 - x), x = 0, and y = 0.

Find the Jacobian, ; x = uv³, y = 8 + uv.

The surface expressed parametrically by x = r cos $\theta$ , y = r sin $\theta$ , z = 90 - r² is

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 = 7xy.

The vector normal to the surface given by , , and when and is

Use spherical coordinates to find the mass of the solid bounded below by and above by if its density is given by .

Find the centroid of the lamina enclosed by x = 5(4y - y²)and the y-axis.

Find the surface area of the portion of the cone that is above the region in the first quadrant bounded by the line , and the parabola .

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 , if and