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

Answer:for 1 ) Normal (rectangular) coordinatesfor 2) Polar coordinatesfor 3) Cylindrical coordinatesfor 4) Spherical coordinates for 5) Normal (rectangular) coordinatesStep-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 dθ3. ∫∫∫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 dθ4. ∫∫∫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 dθ5. ∫∫∫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