evaluate the double integral and reverse order of integration
[(first integral 0 to 1)(second integral 9y to 9)e^(x^2)dx)dy
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.
To evaluate the double integral and reverse the order of integration, we can follow these steps:
Step 1: First, let's draw the region of integration. The given limits of integration suggest that the region lies between x = 0 and x = 1 and between y = 9y and y = 9.
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Step 2: Now, let's set up the double integral with the given limits of integration:
∫∫e^(x^2) dA,
where dA represents the area element.
Step 3: To reverse the order of integration, we need to rewrite the limits of integration based on the new order. In this case, the new order will be dydx.
The inner integral will be with respect to x, and the limits of integration for x will be determined by the region R. In this case, the limits for x are from the lower boundary to the upper boundary of the region, which are x = 9y and x = 9.
The outer integral will be with respect to y, and the limits of integration for y are determined by the region R. In this case, the limits for y are from the left boundary to the right boundary of the region, which are y = 0 and y = 1.
So, the reversed order of integration is:
∫∫e^(x^2) dA = ∫[0 to 1]∫[9y to 9]e^(x^2) dx dy.
Step 4: Finally, we can evaluate the double integral by integrating with respect to x first, then with respect to y second.
∫[0 to 1]∫[9y to 9]e^(x^2) dx dy.
Let's calculate the integral with respect to x:
∫[0 to 1]e^(x^2) dx.
The integral of e^(x^2) does not have a simple closed-form solution, so it cannot be evaluated in terms of elementary functions. However, it can be approximated numerically using methods like the Simpson's rule or the trapezoidal rule.
Once the inner integral is evaluated, we can integrate the result with respect to y over the range [9y to 9].
After performing these calculations, you will obtain the numerical value of the double integral.