Final answer:
To show that the given triple integral is independent of path, we need to evaluate the integral along different paths and show that the results are the same. Perform the integral separately, starting from the innermost integral and moving outward. Substitute the limits of integration and perform the integration to get the final result.
Step-by-step explanation:
To show that the given integral is independent of path, we need to evaluate the integral along two different paths and show that the results are the same. Let's evaluate the integral along the x-axis path first:
∫z=1 to 2 ∫y=3 to 4 ∫x=1 to 2 (2x+1) dx (3y^2) dy (z) dz
Simplifying this integral will give us a numerical value. Now, let's evaluate the integral along the y-axis path:
∫z=1 to 2 ∫y=2 to 3 ∫x=1 to 2 (2x+1) dx (3y^2) dy (z) dz
Again, simplify this integral and you will get a numerical value. If both values are the same, then the integral is independent of path.
To evaluate the integral, perform each integral separately, starting from the innermost integral and moving outward. Substitute the limits of integration and perform the integration to get the final result.
For example, after evaluating the innermost integral, you will end up with an expression involving only y and z. Then, evaluate the y integral using the limits of integration and perform the integration. Finally, evaluate the z integral using the limits of integration and perform the integration. The final result will be the value of the integral.