Chapter 16

Double Integrals

100

∫₁³ ∫₂⁴ (9x³y²) dy dx

3360

200

∫₀¹ ∫_x^2  ^x  (x+3) dy dx

7/12

300

₁∫³  ₀∫^1/y xy² dxdy

1

Triple Integrals

100

∫₀¹ ∫₀^x ∫₀^xy (xyz) dz dy dx 

1/64

200

∫₁⁵ ∫₀^z ∫₂^z (2y) dx dy dz

220/3 or 73 and 1/3

300

∫₀¹ ∫₀^{z^2} ∫₀³   y cos (z⁵) dx dy dz

(3/10) sin (1)

Density, Center of Mass, and Inertia

100

Determine the mass of the lamina that occupies D

D has the bounds y=^√x, y=0, x=0, and x=1; p(x,y) = y

1/4

200

Determine the the moment of inertia I_x of the lamina that occupies D

D has the bounds y=^√x, y=0, x=0, and x=1; p(x,y) = y

1/12

300

Determine the center of mass of the lamina that occupies D

D has the bounds y=^√x, y=0, x=0, and x=1; p(x,y) = y

(2/3,1/30)

Polar, Cylindrical, and Spherical Integrals

100

∫_{π /4}^{π /2}  ∫₀^2√2  (r²) ) dr dθ 

(4√2 π )/3

200

Change to Cylindrical Coordinates

∫_-3³  ∫_{-√(9−x^2)} ^{√(9−x^2)}  ∫_{√(x^2 + y^2)}³   (x² + y²) dz dy dx

∫₀^2π   ∫₀³  ∫_r³  r³  dz dr dθ

300

Convert into Spherical Coordinates

∫∫_Q∫ 1/√ (x²+y²+z²) dV

Q is the sphere x²+y²+z²=25

∫₀ ^2π ∫₀^π   ∫₀⁵  (1/ρ)