Exam 17: Line and Surface Integrals

Find where is the path from to and

Compute the mass of the curve if the mass density is and are measured in centimeters.

Evaluate , where is the rectangle in with vertices at , and , in this order.

Compute the integral where and S is the surface oriented outward.

Compute where S is the part of the surface between the planes and

Let be the triangle with vertices at the points and in counterclockwise order. Compute the line integral .

Find the area of the ellipse cut from the plane by the cylinder .

Evaluate where and oriented upward.

Define and Compute

Evaluate where is the portion of the paraboloid contained within the cylinder

Evaluate where and is the surface oriented outward.

Consider the vector field Find the formula for so that is conservative and

Compute the area of the surface .

Compute the flux of water through the parabolic cylinder (oriented in the positive y-direction) S : , , where the velocity vector is (in ).

Parametrize the surface first using Cartesian coordinates and then using cylindrical coordinates. Assume x and y take on all real values.

Compute the mass of the curve if the mass density is

Let . Compute where C is the spiral

Compute where S is the part of the paraboloid above the -plane oriented upward and

The surface integral where is the part of the plane which lies inside the cylinder is which of the following?

Let . Express in terms of the unit radial vector in .