valuate ∬(x^2+y^2)^2/(x^2y^2) dx dy over the region common to the circles x^2+y62=7x and x^2+y^2=11y
To evaluate the given double integral, we start by finding the limits of integration for both x and y.
Let's start by finding the limits of integration for x.
The first circle equation is x^2 + y^2 = 7x, which we can rewrite as x^2 - 7x + y^2 = 0.
Completing the square for x, we get (x - 7/2)^2 + y^2 = 49/4.
This equation represents a circle with center (7/2, 0) and radius sqrt(49/4) = 7/2.
Now, we need to find the points of intersection between the two circles, which will give us the limits for x.
The second circle equation is x^2 + y^2 = 11y, which we can rewrite as x^2 - 11y + y^2 = 0.
Completing the square for y, we get x^2 + (y - 11/2)^2 = 121/4.
This equation represents a circle with center (0, 11/2) and radius sqrt(121/4) = 11/2.
To find the points of intersection, we can set the two circle equations equal to each other:
(x - 7/2)^2 + y^2 = x^2 + (y - 11/2)^2.
Expanding and simplifying, we get:
x^2 - 7x + (7/2)^2 + y^2 = x^2 - 11y + (11/2)^2.
Rearranging and simplifying further, we have:
-7x + 49/4 = -11y + 121/4.
Multiplying through by 4, we get:
-28x + 49 = -44y + 121.
Rearranging again, we have:
44y - 28x = 72.
Dividing through by 4, we get:
11y - 7x = 18.
Now, we can solve the system of equations comprised of the two circle equations:
x^2 - 7x + y^2 = 0,
11y - 7x = 18.
Rearranging the second equation, we get:
7x + 11y = 18.
Now, we can solve this system of equations to find the points of intersection.
Multiplying the second equation by 7 and subtracting from the first equation, we get:
x^2 - 49 - 11y^2 + 49 = 0.
Simplifying, we have:
x^2 - 11y^2 = 0.
Factoring, we get:
(x - sqrt(11)y)(x + sqrt(11)y) = 0.
So, x = sqrt(11)y or x = -sqrt(11)y.
Substituting these into the equation 7x + 11y = 18, we get:
7(sqrt(11)y) + 11y = 18,
or
7(-sqrt(11)y) + 11y = 18.
Simplifying these equations, we get:
(7 + 11sqrt(11))y = 18,
or
(7 - 11sqrt(11))y = 18.
Solving for y in both equations, we obtain:
y = 18/(7 + 11sqrt(11)),
or
y = 18/(7 - 11sqrt(11)).
Now that we have obtained the y-limits of integration, we need to find the corresponding x-limits.
Substituting the values of y into the equation 7x + 11y = 18, we get:
x = (18 - 11y)/7.
Substituting the value of y = 18/(7 + 11sqrt(11)), we can find the x-coordinate for one of the points of intersection, which gives us the lower limit of integration in x.
Similarly, substituting the value of y = 18/(7 - 11sqrt(11)), we can find the x-coordinate for the other point of intersection, which gives us the upper limit of integration in x.
Now that we have determined the limits of integration for both x and y, we can proceed to evaluate the given double integral:
∬[(x^2 + y^2)^2]/[x^2y^2] dx dy.
Since the limits of integration are quite complicated, it would be challenging to evaluate this integral analytically. However, we can use numerical methods or computer software to find the numerical value of the integral.