Question

Express the integral ?Ef(x,y,z)dV as an iterated integral in six different ways, where E is the solid bounded by z=0,z=7y and x2=49?y

Answer 1

To express the integral ?Ef(x,y,z)dV as an iterated integral in six different ways, the limits of integration for each variable need to be determined. There are six different ways to express the iterated integral, each with different limits for x, y, and z.

Step-by-step explanation:

To express the integral ?Ef(x,y,z)dV as an iterated integral in six different ways, we need to determine the limits of integration for each variable.

1. The first way to express the integral is by using the limits of integration: x goes from -7y to 7y, y goes from 0 to 1, and z goes from 0 to 7y.

2. The second way is by using the limits: y goes from 0 to 1, z goes from 0 to 7y, and x goes from -√(49y) to √(49y).

3. The third way is by using the limits: y goes from 0 to 1, z goes from 0 to 7y, and x goes from -√(49-7y) to √(49-7y).

4. The fourth way is by using the limits: y goes from 0 to 1, x goes from -√(49-7y) to √(49-7y), and z goes from 0 to 7y.

5. The fifth way is by using the limits: y goes from 0 to 1, x goes from -√(49-7y) to √(49-7y), and z goes from 0 to 7x.

6. The sixth way is by using the limits: y goes from 0 to 1, z goes from 0 to 7x, and x goes from -√(49-7y) to √(49-7y).

Answer 2

The question involves setting up a triple integral over a solid with given boundaries as six different iterated integrals by permuting the integration order.

Step-by-step explanation:

The question involves expressing a triple integral over a solid E as an iterated integral in six different ways. Given the boundaries z=0, z=7y, and x^2=49y, we must find the limits for each of the integrals with the correct ordering of the variables x, y, and z. We identify these limits by analyzing the equations:

• 
  For z=0 and z=7y, z ranges from 0 to 7y.
• 
  For x^2=49y, solving for y gives y=x^2/49 and x ranges from -7 to 7.
• 
  For y, once x is chosen, y ranges from 0 to x^2/49.

The iterated integrals can be expressed in six different ways by permuting the order of integration. For example:

1. 
   ∫dx ∫dy ∫dz f(x,y,z), with x from -7 to 7, y from 0 to x^2/49, z from 0 to 7y.
2. 
   ∫dy ∫dx ∫dz f(x,y,z), with y from 0 to 7, x from -√(49y) to √(49y), z from 0 to 7y.
3. 
   ...and so on for the other four permutations.

Each integral is set up differently to account for the bounds of the solid E.