Match the integrals with the type of coordinates which make them the easiest to do. Put the letter of the coordinate system to the left of the number of the integral. 1. ∫10∫y20 1x dx dy 2. ∫∫D 1x2+y2 dA where D is: x2+y2≤4 3. ∫∫∫E z2 dV where E is: - 2≤z≤2, 1≤x2+y2≤2 4. ∫∫∫E dV where E is: x2+y2+z2≤4, x≥0, y≥0, z≥0 5. ∫∫∫E z dV where E is: 1≤x≤2, 3≤y≤4, 5≤z≤6

for 1) Normal (rectangular) coordinates

for 2) Polar coordinates

for 3) Cylindrical coordinates

for 4) Spherical coordinates

for 5) Normal (rectangular) coordinates

Step-by-step explanation:

1. ∫10∫y20 1x dx dy 2. → Normal (rectangular) coordinates x=x, y=y → integration limits ∫ [20,1] and ∫ [10,2]

2. ∫∫D 1x2+y2 dA., D is: x2+y2≤4 → Polar coordinates x=rcosθ, y=rsinθ → integration limits ∫ [2,0] for dr and ∫ [2π,0] for dθ

3. ∫∫∫E z2 dV, E is: - 2≤z≤2, 1≤x2+y2≤2 → Cylindrical coordinates x=rcosθ, y=rsinθ, z=z → integration limits ∫ [2,-2] for dz, ∫ [√2,1] for dr and ∫ [2π,0] for dθ

4. ∫∫∫E dV where E is: x2+y2+z2≤4, x≥0, y≥0, z≥0 → Spherical coordinates x=rcosθcosФ y=rsinθcosФ, z=rsinФ → integration limits ∫ [2,0] for dr,∫ [-π/2,π/2] for dθ, ∫ [π/2,0] for dθ

5. ∫∫∫E z dV where E is: 1≤x≤2, 3≤y≤4, 5≤z≤6 → Normal (rectangular) coordinates x=x, y=y, z=z → integration limits ∫ [2,1] for dx,∫ [4,3] for dy and ∫ [6,5] for dz