1. Use Lagrange multipliers to find the least distance between the point P(?3, 0, 4) and the surface of the cone z 2 = x 2 + y 2 .
2. Evaluate ? 2 0 ? 4 y 2 y sin(x 2 )dxdy.
3. Show that the volume V of the solid in the first octant bounded by the surface z = e y?x , the plane x + y = 1, and the coordinate planes satisfy V < cosh(1)
4. Set up the integral in spherical coordinates ? 1 0 ? ? 1?x2 0 ? ? 2?x2?y 2 ? x2+y 2 xydzdydx
5. Set up in polar coordinates the following double integral ? ? R ? x 2 + y 2dA; (R) bounded by x 2 + y 2 = 2x, y ? x, y ? 0.
6. Suppose a particle moving in space has velocity v(t) = ?sin(2t), cos(t), e?t ? and initial position x(0) = ?2, 1, 0?. Find the position vector function x(t).
7. : Find the maximum value of y x over (x ? 2)2 + y 2 .
1. Use Lagrange multipliers to find the least distance between the point P(?3, 0, 4) and the surface of the cone z 2 = x 2 + y 2 .
2. Evaluate ? 2 0 ? 4 y 2 y sin(x 2 )dxdy.
3. Show that the volume V of the solid in the first octant bounded by the surface z = e y?x , the plane x + y = 1, and the coordinate planes satisfy V < cosh(1)
4. Set up the integral in spherical coordinates ? 1 0 ? ? 1?x2 0 ? ? 2?x2?y 2 ? x2+y 2 xydzdydx
5. Set up in polar coordinates the following double integral ? ? R ? x 2 + y 2dA; (R) bounded by x 2 + y 2 = 2x, y ? x, y ? 0.
6. Suppose a particle moving in space has velocity v(t) = ?sin(2t), cos(t), e?t ? and initial position x(0) = ?2, 1, 0?. Find the position vector function x(t).
7. : Find the maximum value of y x over (x ? 2)2 + y 2 .
