1. Evaluate the given integral by making an appropriate change of variables, where R is the region in the first quadrant bounded by the ellipse 36x^2+25y^2=1. L= double integral R (4sin(144x^2+100y^2) dA.
2. Use the given transformation to evaluate the given integral, where R is the region in the first quadrant bounded by the lines y = x and y = 6x and the hyperbolas xy = 1 and xy = 6. L= double integral R (8xy)dA x=u/v, y=v
3. Evaluate the given integral by making an appropriate change of variables, where R is the trapezoidal region with vertices (9, 0), (10, 0), (0, 10), and (0, 9). L= double integral R (5cos(5y-x/y+x)) dA
1. To evaluate the given integral, we can make the change of variables x = (1/6)cos(theta) and y = (1/5)sin(theta). This will transform the ellipse equation 36x^2 + 25y^2 = 1 into the unit circle equation x^2 + y^2 = 1.
To find the Jacobian of this transformation, we differentiate x and y with respect to theta:
dx/dtheta = (1/6)((-sin(theta))) = -(1/6)sin(theta)
dy/dtheta = (1/5)(cos(theta)) = (1/5)cos(theta)
The Jacobian of the transformation is the determinant of the matrix:
| dx/dtheta dy/dtheta |
| -(1/6)sin(theta) (1/5)cos(theta) |
Jacobian = (1/6)(1/5)sin(theta)cos(theta) + (1/6)(1/5)sin(theta)cos(theta) = 1/30
Now, we can substitute the new variables x and y, and the Jacobian into the integral:
L = double integral (4sin(144x^2 + 100y^2)) dA
= double integral (4sin(144((1/6)cos(theta))^2 + 100((1/5)sin(theta))^2)) * (1/30) d(theta)
= (4/30) double integral sin(24cos^2(theta) + 20sin^2(theta)) d(theta)
Simplifying the expression:
L = (2/15) double integral sin(3cos^2(theta) + 5sin^2(theta)) d(theta)
2. To evaluate the given integral using the given transformation, we substitute x = u/v and y = v.
The lines y = x and y = 6x become v = u and v = 6u, respectively.
The hyperbolas xy = 1 and xy = 6 become xy = u/v = 1 and xy = u/v = 6, respectively.
The region R is bounded by the lines y = x, y = 6x, and the hyperbolas xy = 1 and xy = 6.
To find the Jacobian of the transformation, we differentiate x and y with respect to u and v:
dx/du = 1/v
dx/dv = -u/v^2
dy/du = v
dy/dv = 1/u
The Jacobian of the transformation is the determinant of the matrix:
| dx/du dx/dv |
| dy/du dy/dv |
Jacobian = (1/v)(1/u) - (-u/v^2)(v) = 1/u
Now, we can substitute the new variables x = u/v, y = v, and the Jacobian into the integral:
L = double integral (8xy) dA
= double integral (8(u/v)v) * (1/u) d(uv)
= 8 double integral d(uv)
= 8 integral u from 1 to 6, v from 1 to 6u d(u)
Simplifying the expression:
L = 8 integral (1 to 6) u du
= 8 * (1/2)(6^2 - 1^2)
= 8 * (1/2)(35)
= 140
Therefore, the value of the integral is 140.
3. To evaluate the given integral by making an appropriate change of variables, let's consider the trapezoidal region with vertices (9, 0), (10, 0), (0, 10), and (0, 9).
Let u = 10 - x and v = y + x. We can rewrite the equations of the vertices in terms of u and v:
(9, 0) -> (1, 9)
(10, 0) -> (0, 10)
(0, 10) -> (10, 20)
(0, 9) -> (9, 9)
In this new coordinate system, the region R is a rectangle with vertices (1, 9), (0, 10), (10, 20), and (9, 9).
To find the Jacobian of the transformation, we differentiate x and y with respect to u and v:
dx/du = -1
dx/dv = 1
dy/du = 1
dy/dv = 1
The Jacobian of the transformation is the determinant of the matrix:
| dx/du dx/dv |
| dy/du dy/dv |
Jacobian = (-1)(1) - (1)(1) = -2
Now, we can substitute the new variables u and v, and the Jacobian into the integral:
L = double integral (5cos(5y - x/y + x)) dA
= double integral (5cos(5(v - u)/(v + u) + u)) * (-2) d(uv)
= -10 double integral cos(5(v - u)/(v + u) + u) d(uv)
Simplifying the expression:
L = -10 integral (1 to 9) integral (0 to 10) cos(5(v - u)/(v + u) + u) dv du
Further evaluation of this integral will require the use of numerical methods or specialized software.