MATA33H3 Lecture Notes - Lecture 23: Iterated Integral

Evaluate the surface integral f(x,y,z) dS using an explicit representation of the surface. f(x,y,z) = e , s is the plane z= 10-x-5y in the first octant Set up the surface integral for the given function f(x,y,z) over the given surface S. Integrate with respect to x first. Use increasing limits of integration. dx dy (Type exact answers.)

Find the volume of the solid in the first octant that is bounded by the plane y + z = 4, and the y = x2, and the xy and yz planes. Let u rightarrow(t) = and v rightarrow(t) =. Find d/dt[u rightarrow(t) times v rightarrow(t)]. For r rightarrow(t) =, find the tangent and normal components of acceleration. Find the local maximum and minimum values and saddle points of the function z = f(x, y) = 3x3 + y2 - 9x + 4y. Let S be the surface defined by z = f(x, y) = 1 -y - x2. Let V be the volume of the 3-D region in the first octant bounded by S and the coordinate planes. Set up (but do not evaluate) the iterated integrals for V in two ways: Integrate first respect to x and then respect to y. Integrate first respect to y and then respect to x. Compute the average value of f(x, y) = x2 + y2 over R = {(x, y) | 0 x 2, 0 y 2}. Optimize the function f(x, y, z) = 8x - 4z subject to the constraint x2 + 10y2 + z2 = 5.

Evaluate the surface integral J J f«.yz) dS using an explcit repesentation of the surface. f(x,y,z) = 2xy; s is the plane z = 4-x-y in the first octant. Set up the surface integral for the given function f(x.y,z) over the given surface S. Integrate with respect to x first. Use increasing limits of integration Type exact answers.)