Evaluate ∬(x^2+y^2)^2/(x^2y^2) dx dy over the region common to the circles x^2+y^2=7x and x^2+y^2=11y.
To evaluate the given double integral, we first need to find the region of integration.
The equations x^2 + y^2 = 7x and x^2 + y^2 = 11y represent two circles. In order to find their intersection points, we can solve these equations simultaneously.
Let's start with the equation x^2 + y^2 = 7x:
Rearranging, we get x^2 - 7x + y^2 = 0.
Now, consider the equation x^2 + y^2 = 11y:
Rearranging, we get x^2 + y^2 - 11y = 0.
Since x^2 + y^2 is common to both equations, we can set these equations equal to each other:
x^2 - 7x + y^2 = x^2 + y^2 - 11y.
Simplifying, we have:
-7x = -11y.
Dividing both sides by -7, we get:
x = (11/7)y.
Now, let's substitute this expression for x in one of the original equations, such as x^2 + y^2 = 7x:
(11/7)y^2 + y^2 = 7(11/7)y.
Simplifying further, we have:
(11y^2)/7 + y^2 = (77y)/7.
Now, we can simplify this equation to find the region of integration:
11y^2 + 7y^2 - 77y = 0.
18y^2 - 77y = 0.
y(18y - 77) = 0.
So, we have two possible solutions for y:
1. y = 0.
2. 18y - 77 = 0, which implies y = 77/18.
Now, let's find the corresponding x values for these y values.
For y = 0, using the equation x = (11/7)y, we have:
x = (11/7)(0) = 0.
For y = 77/18, using the equation x = (11/7)y, we have:
x = (11/7)(77/18) = 77/18.
Thus, we have two intersection points: (0, 0) and (77/18, 77/18).
Since the region of integration is common to the circles x^2 + y^2 = 7x and x^2 + y^2 = 11y, it is bounded by these two circles and the line y = x.
To evaluate the given double integral, we can set up the integral using polar coordinates. In polar coordinates, x = rcosθ and y = rsinθ.
The limits of integration for r will be the radii of the circles, which are 7 (from the first equation) and 11 (from the second equation).
The limits of integration for θ will be the angles that sweep out the region of integration between the two circles and the line y = x. To find these limits, we need to determine the angle at each intersection point.
At the point (0, 0), the angle is 0, and at the point (77/18, 77/18), the angle can be found using the equation tanθ = y/x. So, we have tanθ = (77/18)/(77/18) = 1, which gives us θ = π/4.
Now, we can set up the double integral:
∬(x^2 + y^2)^2 / (x^2y^2) dx dy
In polar coordinates, this becomes:
∬(r^2)^2 / ((r^2cosθ)^2(r^2sinθ)^2) r dr dθ
Simplifying further, we have:
∬r^2 / (r^4cos^2θsin^2θ) r dr dθ
Now, we can evaluate this integral over the limits of integration: r from 7 to 11, and θ from 0 to π/4.
∫[θ=0 to π/4] ∫[r=7 to 11] (r^2) / (r^4cos^2θsin^2θ) r dr dθ
Evaluating this integral will give the final result.