Find the area in the first quadrant bounded by the arc of the circle described by the polar equation
r = (2 sin theta)+(4 cos theta)
A. 5pi/2
B. (5pi/2)+4
C. 5pi
D. 5pi + 8
Recall that in polar coordinates,
A = Int(1/2 r^2 dθ)
r^2 = 4 sin^2θ + 8sinθcosθ + 16cos^2θ
= 4 + 4sin2θ + 12cos^2θ
= 10 + 4sin2θ + 6cos2θ
A = Int(5 + 2sin2θ + 3cos2θ)dθ [0,pi/2]
= 5t - cos2θ + 3/2 sin2θ [0,pi/2]
= [5pi/2 + 1 + 0] - [0 - 1 + 0]
= 5pi/2 + 2
To find the area bounded by the arc of the circle described by the polar equation r = (2 sin theta) + (4 cos theta), we can use the following steps:
Step 1: Convert the polar equation to rectangular coordinates.
To convert the polar equation to rectangular coordinates, we can use the formulas:
x = r cos(theta)
y = r sin(theta)
Substituting the given equation, we have:
x = (2 sin theta) + (4 cos theta) cos(theta)
y = (2 sin theta) + (4 cos theta) sin(theta)
Step 2: Determine the points of intersection.
To find the points of intersection, we need to solve the equations for x and y equal to zero simultaneously.
Setting x = 0, we have:
0 = (2 sin theta) + (4 cos theta) cos(theta)
Dividing both sides by cos(theta), we get:
0 = 2 tan(theta) + 4
Simplifying, we have:
tan(theta) = -2
Setting y = 0, we have:
0 = (2 sin theta) + (4 cos theta) sin(theta)
Dividing both sides by sin(theta), we get:
0 = -4 + 2 cot(theta)
Simplifying, we have:
cot(theta) = 2
Step 3: Find the values of theta.
To find the values of theta, we can use the trigonometric identities:
tan(theta) = sin(theta) / cos(theta)
cot(theta) = cos(theta) / sin(theta)
Using these identities, we can rewrite the equations as:
sin(theta) / cos(theta) = -2
cos(theta) / sin(theta) = 2
Simplifying, we get:
sin(theta) = -2 cos(theta)
cos(theta) = 2 sin(theta)
Dividing the second equation by cos(theta), we have:
tan(theta) = 2
Therefore, theta can be found by taking the inverse tangent of -2 and 2:
theta = arctan(-2) and theta = arctan(2)
Step 4: Calculate the area.
The area bounded by the arc of the circle is given by the integral of 1/2 * r^2 d(theta) from theta = arctan(-2) to theta = arctan(2).
Substituting the polar equation, we have:
Area = 1/2 * integral((2 sin theta + 4 cos theta)^2) d(theta) from arctan(-2) to arctan(2)
Simplifying the integral, we can expand the square and distribute:
Area = 1/2 * integral(4 sin^2 theta + 16 sin theta cos theta + 16 cos^2 theta) d(theta) from arctan(-2) to arctan(2)
Using trigonometric identities, we know that:
sin^2 theta = (1 - cos(2 theta))/2
cos^2 theta = (1 + cos(2 theta))/2
sin theta cos theta = (sin(2 theta))/2
Substituting these identities into the integral, we have:
Area = 1/2 * integral(4(1 - cos(2 theta))/2 + 16(sin(2 theta))/2 + 16(1 + cos(2 theta))/2) d(theta) from arctan(-2) to arctan(2)
Simplifying the integral, we have:
Area = 1/2 * integral(4 - 4 cos(2 theta) + 8(sin(2 theta)) + 8 + 8 cos(2 theta)) d(theta) from arctan(-2) to arctan(2)
Combining like terms, we have:
Area = 1/2 * integral(20 + 4 sin(2 theta)) d(theta) from arctan(-2) to arctan(2)
Integrating, we get:
Area = [20 theta - 2 cos(2 theta)]/2 from arctan(-2) to arctan(2)
Now, we substitute the values of theta:
Area = [20 arctan(2) - 2 cos(2 arctan(2))]/2 - [20 arctan(-2) - 2 cos(2 arctan(-2))]/2
Simplifying further, we can use the values of arctan(2) and arctan(-2):
Area = [20 arctan(2) - 2 (3/5)]/2 - [20 arctan(-2) - 2 (-3/5)]/2
Therefore, the exact area bounded by the arc of the circle described by the polar equation r = (2 sin theta) + (4 cos theta) is given by:
Area = [20 arctan(2) - 2 (3/5)]/2 - [20 arctan(-2) - 2 (-3/5)]/2
This cannot be simplified further, so the exact area is the final answer.
To find the area in the first quadrant bounded by the given polar equation, you need to integrate the function from the initial value of theta to the final value of theta.
First, let's find the points where the curve intersects the x-axis, which corresponds to the values of theta where r = 0.
Setting r = 0, we have:
0 = 2 sin theta + 4 cos theta
Dividing both sides by 2, we get:
sin theta + 2 cos theta = 0
Rearranging terms, we have:
sin theta = -2 cos theta
Dividing both sides by cos theta:
tan theta = -2
Now, find the values of theta in the first quadrant (0 to π/2) where tan theta = -2.
To do this, calculate the arctan(-2):
theta = arctan(-2)
Using a calculator, you will find that theta is approximately -63.43 degrees.
However, since we are only interested in the values of theta in the first quadrant, we need to find the reference angle. The reference angle is the positive value that, when added to or subtracted from 180 degrees, gives us the negative value of -63.43 degrees.
Reference angle = 180 - 63.43 = 116.57 degrees
Converting this angle to radians, we have:
Reference angle = 116.57 degrees * (π/180) = 2.04 radians
Now we have the initial value of theta as 0 and the final value as 2.04 radians.
To find the area, we need to integrate the function r with respect to theta over this range:
A = ∫[0 to 2.04] (1/2) r^2 dθ
Substituting the given equation for r, we have:
A = ∫[0 to 2.04] (1/2)((2 sin θ) + (4 cos θ))^2 dθ
Expanding and simplifying the expression inside the integral, we get:
A = ∫[0 to 2.04] (1/2)(4 sin^2 θ + 16 cos^2 θ + 16 sin θ cos θ) dθ
Using trigonometric identities (sin^2 θ = 1 - cos^2 θ), we can rewrite the integral as:
A = ∫[0 to 2.04] (1/2)(4 - 4 cos^2 θ + 16 cos^2 θ + 16 sin θ cos θ) dθ
Simplifying further, we have:
A = ∫[0 to 2.04] (1/2)(20 cos^2 θ + 16 sin θ cos θ) dθ
Now, we can integrate each term separately:
A = (1/2) ∫[0 to 2.04] (20 cos^2 θ + 16 sin θ cos θ) dθ
The integral of cos^2 θ is (1/2)(θ + sin 2θ), and the integral of sin θ cos θ is (1/2)sin^2 θ.
Substituting these results back into the equation:
A = (1/2)[(20θ + 20 sin 2θ) / 2 + 16(sin^2 θ / 2)][0 to 2.04]
Simplifying further, we get:
A = (1/4)[(20θ + 20 sin 2θ) + 16(sin^2 θ)][0 to 2.04]
Evaluating this expression at the upper limit (2.04 rad) and subtracting it from the evaluation at the lower limit (0 rad), we can find the area.
Calculating this, we get:
A = (1/4)[(20(2.04) + 20 sin 2(2.04)) + 16(sin^2 2.04) - (20(0) + 20 sin 2(0)) - 16(sin^2 0)]
Simplifying, we find that the area in the first quadrant bounded by the given curve is approximately 5.13 square units.
Therefore, the correct option is not listed among the given choices.