Multivariate Calculus Review 2

This topic is a collection of problems and concepts that might help someone understand their working knowledge of multivariate calculus.

(1) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(2) Find the critical points and classify each as a relative maximum, a relative minimum, or a saddle point for the function

(3) Find the absolute maximum and minimum values of the function on the triangular region with vertices and

(4) Find the absolute maximum and minimum values of the function on the square region with vertices and

(5) A rectangular box with no top is to have a fixed volume. What should its dimensions be if we want to use the least amount of material in its construction?

(6) Use the method of Lagrange multipliers to maximize the function subject to

(7) Use the method of Lagrange multipliers to minimize the function subject to

(8) Use the method of Lagrange multipliers with two parameters to minimize subject to and

(9) Evaluate the double integral:

    (a)

    (b)

(10) Find the volume of the solid region bounded below by the given rectangle in the -plane and above by the graph of the given surface.

    (a) on

    (b) on

(11) If is a constant function, say and show that

(12) Show that the iterated integrals

and

have different values.

(13) Sketch the region and compute the iterated integral over the non-rectangular region given

(14) Sketch the region and compute the iterated integral over the non-rectangular region given multivariate calculus review 2 _gr_35.gif]

(15) Sketch the region and compute the iterated integral over the non-rectangular region given

(16) Sketch the region and compute the iterated integral over the non-rectangular region given

(17) Sketch the region and compute the iterated integral over the non-rectangular region given

(18) Sketch the region and compute the iterated integral over the non-rectangular region given

(19) Find the volume of the solid bounded by (Setup but do not evaluate the double integral)

(20) Find the volume of the solid that lies inside both the cylinder and the sphere . (Setup but do not evaluate the double integral).

(21) Use polar coordinates to evaluate the iterated integrals multivariate calculus review 2 _gr_43.gif]

(22) Use polar coordinates to evaluate the iterated integrals multivariate calculus review 2 _gr_44.gif]

(23) Find the volume of the solid region common to the cylinder and the ellipsoid

(24) Find the volume of the solid region bounded above by the cone below by the plane and on both sides by the cylinder

(25) Find the surface area of the surface of the portion of the sphere inside the cylinder

(26) Find the surface area of the surface of the portion of the cone inside the cylinder

(27) Find the area of the surface given parametrically by the equation

for

(28) Find the volume of the solid bounded by the sphere and the paraboloid

(29) Find the volume of the solid of the region bounded by the cylinders and and the planes and

(30) Find the volume of the region between the two elliptic paraboloids and

(31) Change the order of integration to show that

Also, show that

(32) Higher-dimensional multiple integrals can be defined and evaluate in essentially the same way as double integrals and triple integrals. Evaluate the multiple integrals where is the four-dimensional "hyperbox" defined by and

(33) Find the centroid for a lamina with over the region bounded by the curve and the line in the first octant.

(34) Use double integration to find the center of mass of a lamina covering the region bounded by and with density function

(35) A lamina has the shape of a semicircular region Find the center of mass of the lamina if the density at each point is directly proportional to the square of the distance from the point to the origin.