To evaluate the double integral ∫∫e^(x^2) dA, where the first integral is from 0 to 1 and the second integral is from 9y to 9, with respect to x and y, we can proceed as follows:
Step 1: Reverse the order of integration.
To reverse the order of integration, we need to redraw the region of integration in the xy-plane. The given limits indicate that the region lies between the lines x = 0 and x = 1, and above the curve y = 9x/9.
Step 2: Determine the new limits of integration.
The limits of integration for y will be determined by the lines x = 0 and x = 1. Since the curve y = 9x/9 is a straight line, we can determine the y-limits by substituting x = 0 and x = 1 into the equation.
When x = 0: y = 9(0)/9 = 0
When x = 1: y = 9(1)/9 = 1
Therefore, the y-limits of integration are from y = 0 to y = 1.
The limits of integration for x will be determined by the curve y = 9x/9. To find the x-limits, we solve the equation for x.
y = 9x/9
x = y
Therefore, the x-limits of integration are from x = 0 to x = y.
Step 3: Evaluate the integral.
The reversed integral becomes:
∫∫e^(x^2) dA = ∫0^1 ∫0^y e^(x^2) dx dy
Now we can evaluate the integral by integrating with respect to x first, and then with respect to y.
∫0^1 e^(x^2) dx = [e^(x^2)] evaluated from 0 to 1
= e^(1^2) - e^(0^2)
= e - 1
Next, we integrate the above result with respect to y.
∫0^1 (e - 1) dy = (e - 1) ∫0^1 dy
= (e - 1) [y] evaluated from 0 to 1
= (e - 1)(1 - 0)
= e - 1
Therefore, the value of the double integral ∫∫e^(x^2) dA with reversed order of integration is e - 1.