1) Compute the sarface area S of the sphere x²+ y²+ z²= a². 4πa²
2) Find the area of the surface of the sphere x²+ y²+ z²= a² cut off by x²+ y²= ax. 2(π - 2)a²
3) Use Stoke's theorem to find the line integerls ∫ x²y³ dx + dy + z dz, where C is the circle x²+ y²= a², z=0. πa⁶/8. ᶜ
4) Prove that the volume common to the sphere x²+ y²+ z²= a² and the cylinder x²+ y²= ay is 2(3π -4)a³/9
5) Show that the volume common to the cylinder x²+ y²= a² and x²+ z²= a² is 16a³/3.
6) Compute the volume of the ellipsoid x²/a²+ y²/b²+ z²/c²= 1. 4πabc/3
7) Using Gauss's theorem, show that ∫∫ (xz dx dy+ xy dy dz+ yz dz dx)
ˢ
Where S is the pyramid formed by the planes x= 0, y=0, z=0 and x + y + z=1 is 1/8.
8) Evaluate ∫∫∫ dxdydz/(x + y+ z+1)³
ⱽ
Where V is the tetrahedron bounded by the planes x=0, y=0, z=0, x + y+ z=1. (1/16) Log(256/e⁵)
9) Show that ∫∫∫ (x²+ y²+ z²) xyz dx dy dz= 0 where V= {(x, y, z): x²+ y²+ z²≤ 1}.
10) Show that ∫∫∫ dx dy dz/{x²+ y²+ (z -2)²}= π(2 - (3/2) log 3) evaluated over the solid x²+ y²+ z²≤ 1.
11) Evaluate ∫∫∫ ₑ√(x²/a² + y²/b²+ z²/c²) dx dy dz over the ellipsoid x²/a² + y²/b²+ z²/c² ≤ 1 (a, b, c > 0). 4πabc(e -2)
12) Prove ∫∫∫ (x²+y²+z²)dx dy dz = (2/5) πa⁵
ᴱ
Where E is the region bounded by the hemisphere z≥ 0, x²+ y²+ z²= a².
13) Evaluate ∫∫∫ √(1 - x²/a² - y²/b² - z²/c²) dx dy dz
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Where D is the closed region x²/a²+ y²/b²+ z²/c² ≤ 1 (a, b, c > 0). abcπ²/4
14) Find the area of the part of the spherical surface x²+ y²+ z²= 4a² enclosed by the cylinder (x²+ y²)²= 2a²(2x²+ y²).
15) Find the area of the part of y²+ z²= x² inside the cylinder x²+ y²= a².
16) Find the area of the part of z²= 4x cut off by the cylinder y²= 4x and the plane x=1.
17) Find the area of the surface of the cylinder x²+ y²= 4a² above the xy-plane and bounded by the planes y=0, z= a and y= z.
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