How do I go about this?
I need details solution to study
find any given rectangle ∆ R=(xbase0,xbase0+w).(ybase0.ybase0+h) if we define function A A(R)=
∫ ∫R e^(2iπ(x+y)dxdy
????
what are the range of x,y. All real positive numbers?
Sir I don't know I saw the question like that I became so confuse
What do you have in mind sir ,with the range??
To find the area of the given rectangle ∆ R, we need to evaluate the double integral of the function A(R) = ∫ ∫R e^(2iπ(x+y))dxdy over the rectangle R defined by R = (xbase0, xbase0 + w).(ybase0, ybase0 + h). Here's how you can go about solving this:
1. Calculate the integral with respect to x:
∫R e^(2iπ(x+y)) dx = [1/(2iπ)] e^(2iπ(x+y)) + C1
Note: C1 represents a constant of integration that would be determined later.
2. Substitute the upper and lower limits of x into the integral from step 1:
[1/(2iπ)] e^(2iπ(x+y)) + C1 |xbase0 to (xbase0+w)
Applying the limits, we get:
[1/(2iπ)] [e^(2iπ(xbase0+y)) - e^(2iπ(xbase0+w+y))] + C1
3. Apply the same process to integrate with respect to y:
∫R [1/(2iπ)] [e^(2iπ(xbase0+y)) - e^(2iπ(xbase0+w+y))] + C1 dy
Simplifying the integral, we get:
[1/(2iπ)] [ (e^(2iπ(xbase0+y)) - e^(2iπ(xbase0+w+y)))y + C1y ] + C2
Note: C2 represents a new constant of integration that would be determined later.
4. Substitute the upper and lower limits of y into the integral from step 3:
[1/(2iπ)] [ (e^(2iπ(xbase0+y)) - e^(2iπ(xbase0+w+y)))y + C1y ] + C2
|ybase0 to (ybase0+h)
Applying the limits, we get:
[1/(2iπ)] [ (e^(2iπ(xbase0+y)) - e^(2iπ(xbase0+w+y)))y + C1y ] + C2
|ybase0 to (ybase0+h)
5. Simplify the expression by evaluating the limits:
[1/(2iπ)] [ (e^(2iπ(xbase0+y)) - e^(2iπ(xbase0+w+y)))y + C1y ] + C2
|ybase0 to (ybase0+h)
After substituting the lower and upper limits, simplify the expression further if applicable.
6. Finally, simplify and solve the expression to find the area of the given rectangle.
It's important to note that this process assumes that all calculations are performed correctly and the function A(R) is well-defined. Additionally, please double-check the given rectangle definition and the function to ensure accuracy while solving the integral.