The region R, is bounded by the graphs of x = 5/3 y and the curve C given by x = (1+y^2)^(1/2), and the x-axis.
a) Set up and evaluate an integral expression with respect to y that gives the area of R.
b) Curve C is part of the curve x^2 - y^2 = 1, Show that x^2 - y^2 = 1 can be written as the polar equation r^2 = 1/(cos^2è - sin^2è).
c) Use the polar equation given in part b) to set up an integral expression with respect to theta that represents the area of R.
a) To find the area of region R using an integral expression with respect to y, we need to determine the limits of integration and the integrand.
First, let's find the y-values where the two curves intersect to determine the limits of integration. We set the equations x = 5/3y and x = (1+y^2)^(1/2) equal to each other:
5/3y = (1+y^2)^(1/2)
Squaring both sides:
25/9y^2 = 1+y^2
Subtracting y^2 from both sides and rearranging:
24/9y^2 = 1
Simplifying:
8/3y^2 = 1
Dividing both sides by 8/3:
y^2 = 3/8
Taking the square root of both sides:
y = ±√(3/8) = ±(√3/2)/2
Since the integral expression is with respect to y, these y-values will be the limits of integration. However, we only need the positive y-value since the negative y-value lies below the x-axis (and we are interested in the area above the x-axis).
Next, let's express the area element dA in terms of y. Since the region R is bounded by the x-axis and the curves, we can express the area element as dA = x dy.
We know that x = (1+y^2)^(1/2), so dA = (1+y^2)^(1/2) dy.
Therefore, the integral expression for the area of R is:
Area = ∫[from √(3/8) to 0] (1+y^2)^(1/2) dy
b) We have the equation x^2 - y^2 = 1. To show that this can be written as the polar equation r^2 = 1/(cos^2θ - sin^2θ), we need to transform the Cartesian equation into polar coordinates.
In polar coordinates, we have x = r cosθ and y = r sinθ. Substituting these into the equation x^2 - y^2 = 1:
(r cosθ)^2 - (r sinθ)^2 = 1
Simplifying:
r^2 cos^2θ - r^2 sin^2θ = 1
Factoring out r^2:
r^2 (cos^2θ - sin^2θ) = 1
Dividing both sides by (cos^2θ - sin^2θ):
r^2 = 1/(cos^2θ - sin^2θ)
Therefore, x^2 - y^2 = 1 can indeed be written as the polar equation r^2 = 1/(cos^2θ - sin^2θ).
c) Now that we have the polar equation r^2 = 1/(cos^2θ - sin^2θ), we can use it to set up an integral expression with respect to θ to represent the area of region R.
The area element in polar coordinates is given by dA = (1/2) r^2 dθ.
Therefore, the integral expression for the area of R in polar coordinates is:
Area = (1/2) ∫[from θ1 to θ2] (1/(cos^2θ - sin^2θ)) dθ
Here, θ1 and θ2 are the angles that correspond to the limits of integration determined by the intersection points of the curve C with the x-axis.
Note that to evaluate this integral, you would need to find the precise limits of integration based on the specific intersection points, which were not provided in the question.