evaluate double integral y dydx where r is the region in first quadrant enclosed between the circle x^2+y^2=25 and straight line x+y=25

To solve this double integral, we first need to set up the limits of integration. We are given that the region R is in the first quadrant and is enclosed by the circle x^2 + y^2 = 25 and the line x + y = 25.

Step 1: Convert to Polar Coordinates Since we are dealing with a circle, it is easier to convert to polar coordinates. The equation for the circle in polar coordinates is r^2 = 25, so r = 5. The equation for the line in polar coordinates is rcos(θ) + rsin(θ) = 25, so r = 25/(cos(θ) + sin(θ)).

Step 2: Set up the Limits of Integration The limits of r are from 0 to 5. The limits of θ are from 0 to π/4. This is because the line x + y = 25 intersects the x-axis at x = 25 and the y-axis at y = 25, which corresponds to an angle of π/4 in the first quadrant.

Step 3: Set up the Double Integral The double integral is ∫ (from 0 to π/4) ∫ (from 0 to 5) r*sin(θ) * r dr dθ. The extra r is from the Jacobian determinant when converting to polar coordinates.

Step 4: Solve the Inner Integral ∫ (from 0 to 5) r^2 dr = [r^3/3] (from 0 to 5) = 125/3.

Step 5: Solve the Outer Integral ∫ (from 0 to π/4) (125/3)sin(θ) dθ = [(125/3) (-cos(θ))] (from 0 to π/4) = (125/3) * (1 - sqrt(2)/2).

So, the value of the double integral is (125/3) * (1 - sqrt(2)/2).