Match the integrals with the type of coordinates which make them the easiest to do.
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1. ∭E dV where E is:
x^2 + y^2 + z^2<= 4, x>= 0, y>= 0, z>= 0
2. ∭E z^2 dV where E is:
-2 <= z <= 2,1 <= x^ 2 + y^2 <= 2
3. ∭E z dV where E is:
1 <= x <= 2, 3<= y <= 4,5 <= z <= 6
4. ∫10∫y^20 1/x dx
5. ∬D 1/x^2 + y^2 dA where D is: x^2 + y^2 <=4
Options:
A. spherical coordinates
B. polar coordinates
C. cylindrical coordinates
D. cartesian coordinates

1. C. cylindrical coordinates

2 A. spherical coordinates

3. A. spherical coordinates

4. D. Cartesian coordinates

5  B. polar coordinates

Step-by-step explanation:

USE THE BOUNDARY INTERVALS TO IDENTIFY

1. ∭E dV where E is:  

x^2 + y^2 + z^2<= 4, x>= 0, y>= 0, z>= 0  -- This is A CYLINDRICAL COORDINATES SINCE x>= 0, y>= 0, z>= 0

2. ∭E z^2 dV where E is:  

-2 <= z <= 2,1 <= x^ 2 + y^2 <= 2 This is A SPHERICAL COORDINATES

3. ∭E z dV where E is:

1 <= x <= 2, 3<= y <= 4,5 <= z <= 6 -- This is A SPHERICAL COORDINATES

4. ∫10∫y^20 1/x dx   This is A CARTESIAN COORDINATES

5. ∬D 1/x^2 + y^2 dA where D is: x^2 + y^2 <=4  This is A POLAR COORDINATES

b. astronomers