1.Consider the solid tetrahedronE with vertices

O(0,0,0),A(7,0,0),B(0,7,0), andC(0,0,7)

.

(a) (1 pt.) Sketch the solidE in thexyz-space. Clearly label the axes and all the relevant points onE.

(b) (2 pts.) Find the equation of the plane passing through the pointsA,B, andC.

(c) (2 pts.) LetD be the projection of the plane found in part (b) onto thexy-plane. Draw the regionD in thexy-plane. Clearly label the axes, all lines, curves, and relevant points onD.

(d) (2 pts.) Write the double integral to find the volume of the solidE.

(e) (2 pts.) Write the triple integral to find the volume of the solidE.

(f) (5 pts.) Find the volume of the solidE by evaluating the integral found in part (d) or in part (e).

2.Consider the solidE that lies within the spherex² +y² +z² =64 and above the cone

z =

x2 +y2

.

(a) (2 pts.) Convert the equation of the spherex² +y² +z² =64 into a spherical equation using,, and.

(b) (2 pts.) Convert the equation of the cone

z =

x2 +y2

into a spherical equation using,, and.

(c) (1 pt.) Sketch the solidE in thexyz-space and describe the shape ofE. Clearly label the axes, all surfaces and relevant points onE.

(d) (1 pt.) A triple integral

	f(x,y,z)dV

is set up to find the volume of the solidE using rectangular coordinates. Find the function

f(x,y,z)

.

(e) (3 pts.) Express the integration regionE in spherical coordinates and convert the triple integral found in part (d) into spherical coordinates.

(f) (5 pts.) Find the volume of the solidE by evaluating the triple integral found in (e).

3.Consider the double integral:

	8xdA, whereD is the triangular region with verticesO(0, 0),A(1, 2), andB(0, 3)
D

(a) (1 pt.) Sketch the integration regionD in thexy-plane. Clearly label the axes and all the relevant points onD.

(b) (2 pts.) Let

L₁

be the line connecting the pointsO andA, and let

L₂

be the line connecting the pointsA andB. Find the equations of the lines

L₁

and

L₂

.

(c) (2 pts.) ExpressD as a region of type I. Write the iterated integral using the region. Clearly state the integration order.

(d) (4 pts.) ExpressD as a region of type II. Write the iterated integral using the region. Clearly state the integration order.

(e) (5 pts.) Evaluate the iterated integral found either in part (c) or in part (d).