Use symmetry to evaluate the double integral ∫∫R(10+x^2⋅y^5) dA, R=[0, 6]×[−4, 4].
(Give your answer as an exact number.)
∫∫R(10+x^2⋅y^5) dA=
The integral is easy enough, but I sure don't see where symmetry helps, especially in that region R.
To use symmetry to evaluate the double integral, we need to identify any symmetries in the region of integration and the function being integrated.
In this case, the region of integration R = [0, 6] × [−4, 4] is a rectangle. We can observe that this rectangle is symmetric about the x-axis.
Now, let's consider the function f(x, y) = 10 + x^2⋅y^5. We can see that this function is symmetric about the y-axis because the variable x appears only in terms of x^2.
Based on these observations, we can use symmetry to simplify the double integral:
∫∫R(10+x^2⋅y^5) dA
= 2 ∫∫R_{1st quadrant} (10+x^2⋅y^5) dA
Here, we have divided the region R into four quadrants and chosen the first quadrant because it is the one in which both x and y are positive.
Now, we can evaluate the double integral over the first quadrant using the appropriate limits:
∫∫R_{1st quadrant} (10+x^2⋅y^5) dA
= 2 ∫_{y=0}^{4} ∫_{x=0}^{6} (10+x^2⋅y^5) dx dy
Evaluating this integral will give us the exact answer.