Ind The Area Of The Part Of The Circle R=4sin+Cos In The Fourth Quadrant
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Introduction to the Problem
In the realm of coordinate geometry and polar equations, understanding the area enclosed by specific parts of curves is a fundamental concept. The problem of finding the area of a particular segment or sector of a circle defined by a polar equation is common in advanced mathematics, physics, and engineering applications. In this article, we focus on calculating the area of the part of the circle given by the polar equation R = 4sinθ + cosθ that resides specifically in the fourth quadrant.
This exploration involves translating the polar equation into a more familiar Cartesian form, identifying the relevant angular ranges, and then applying the appropriate integral calculus techniques to compute the area. By the end of this discussion, readers will understand the step-by-step process involved in solving such problems, along with insights into the geometric interpretation of the results.
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Understanding the Polar Equation R = 4sinθ + Cosθ
What Is a Polar Equation?
A polar equation describes a curve in the plane using a radius (R) and an angle (θ). For each value of θ, the radius R specifies how far from the origin the point lies. The equation R = 4sinθ + cosθ combines sine and cosine functions, resulting in a curve that exhibits certain symmetries and properties.
Analyzing R = 4sinθ + Cosθ
This particular equation can be rewritten to better understand its geometric nature:
- Recognize that R = a sinθ + b cosθ can be expressed as R = √(a² + b²) sin(θ + α), where α is an angle satisfying certain conditions.
- For R = 4sinθ + cosθ:
- a = 4
- b = 1
- Compute the amplitude:
√(4² + 1²) = √(16 + 1) = √17 ≈ 4.1231
- Find the phase shift α:
α = arctangent(b / a) = arctangent(1 / 4) ≈ 14.04°
This form indicates that the original curve is a sinusoidally shifted circle or a similar conic, which helps in visualizing and calculating the area.
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Geometric Interpretation of the Curve
Shape and Symmetry
The equation R = 4sinθ + cosθ describes a circle in the polar coordinate system. To confirm this, we can convert it to Cartesian coordinates or analyze its properties:
- Using the identities:
- x = R cosθ
- y = R sinθ
- Substituting R:
- x = (4sinθ + cosθ) cosθ
- y = (4sinθ + cosθ) sinθ
- Alternatively, it might be more straightforward to convert the polar form into Cartesian form directly.
Conversion to Cartesian Coordinates
Starting from the polar equation:
R = 4sinθ + cosθ
Express R in terms of x and y:
- R² = x² + y²
- cosθ = x / R
- sinθ = y / R
Substitute into the original equation:
R = 4(y / R) + (x / R)
Multiply both sides by R:
R² = 4y + x
Replace R² with x² + y²:
x² + y² = 4y + x
Bring all terms to one side:
x² + y² - 4y - x = 0
Complete the square for x and y:
- For x:
x² - x = (x - 0.5)² - 0.25
- For y:
y² - 4y = (y - 2)² - 4
Rewrite the equation:
(x - 0.5)² - 0.25 + (y - 2)² - 4 = 0
Combine constants:
(x - 0.5)² + (y - 2)² = 4.25
This is the equation of a circle with:
- Center at (0.5, 2)
- Radius √4.25 ≈ 2.0616
Visualizing the Circle
The circle's position and size are now clear:
- Located near the point (0.5, 2)
- With a radius of approximately 2.06 units
Understanding the circle's position helps determine which parts lie in the fourth quadrant.
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Identifying the Fourth Quadrant Segment
Definition of the Fourth Quadrant
In Cartesian coordinates, the fourth quadrant is where:
- x > 0
- y < 0
In polar coordinates, the fourth quadrant corresponds to:
- θ between 270° (3π/2) and 360° (2π)
or in radians:
- θ in (3π/2, 2π)
Angular Range for the Part in the Fourth Quadrant
Since the circle is centered at (0.5, 2), which is above the x-axis, and has a radius of approximately 2.06, parts of it will extend into the fourth quadrant depending on its position.
To find the exact angular range:
- Determine the points where the circle intersects the x-axis (y=0), which are critical in defining the segment.
Substitute y=0 into the circle's equation:
(x - 0.5)² + (0 - 2)² = 4.25
Simplify:
(x - 0.5)² + 4 = 4.25
(x - 0.5)² = 0.25
x - 0.5 = ±0.5
x = 0 or x = 1
Corresponding points:
- (0, 0): at the origin
- (1, 0): at x=1, y=0
Now, consider the angles θ corresponding to these points:
- For (0, 0):
r = √(0² + 0²) = 0
θ is undefined, but the point is at the origin.
- For (1, 0):
r = 1
θ = arctangent(0 / 1) = 0°
Since θ=0 corresponds to the positive x-axis, the points of intersection are at:
- (1, 0): θ=0°, r=1
- (0, 0): at the origin, which is the intersection point of the circle and the x-axis.
Given the circle's center at (0.5, 2), and radius approximately 2.06, the circle extends into the fourth quadrant near θ approaching 2π, but precise calculation involves solving for the angles at which the circle intersects the axes.
Calculating the Sector Area
To find the area of the part of the circle lying in the fourth quadrant:
- Identify the angular limits (θ1 and θ2) corresponding to the intersection points of the circle with the axes or with lines y=0 and x>0.
- Determine whether the circle extends into the fourth quadrant based on these points.
- Use the polar area formula:
\[
A = \frac{1}{2} \int{\theta1}^{\theta_2} R^2 \, d\theta
\]
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Computing the Area in the Fourth Quadrant
Step 1: Find the Relevant Angular Limits
Given the circle's Cartesian equation:
\[
(x - 0.5)^2 + (y - 2)^2 = 4.25
\]
Expressed in polar coordinates:
\[
x = R \cos\theta \\
y = R \sin\theta
\]
Substitute into the circle's equation:
\[
(R \cos\theta - 0.5)^2 + (R \sin\theta - 2)^2 = 4.25
\]
Expanding:
\[
R^2 \cos^2\theta - R \cos\theta + 0.25 + R^2 \sin^2\theta - 4 R \sin\theta + 4 = 4.25
\]
Combine like terms:
\[
R^2 (\cos^2\theta + \sin^2\theta) - R (\cos\theta + 4 \sin\theta) + (0.25 + 4) = 4.25
\]
Since \(\cos^2\theta + \sin^2\theta = 1\):
\[
R^2 - R (\cos\theta + 4 \sin\theta) + 4.25 = 4.25
\]
Subtract 4.25 from both sides:
\[
R^2 - R (\cos\theta + 4 \sin\theta) = 0
\]
Factor:
\[
R [ R - (\cos\theta + 4 \sin\theta) ] = 0
\]
Thus, either:
- R = 0, which corresponds to the origin, or
- \( R = \cos\theta + 4 \sin\theta \)
Recall that R = 4 sinθ + cosθ, so the second solution matches the original equation, confirming the points on the circle.
Step 2: Find the θ Values at the Intersection Points
At the intersection with the axes, the radius R is zero:
\[
0 = 4 \sin
