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.