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Magazine Chapter 10: Problem 14
Use a graphing utility to graph the polar equation and find the area of the given region.Inner loop of \(r=4-6 \sin \theta\)
Short Answer
Expert verified
Firstly, understand the polar function and sketch the graph. Identify the inner loop, and use the area formula for polar coordinates to set up the integral. Subsequently compute the integral to obtain the area.
Step by step solution
01
Understand the polar function: \(r=4-6 \sin \theta\)
The graph of any function of the form \(r = a \pm b \sin \theta \) or \(r = a \pm b \cos \theta \) (where both 'a' and 'b' are positive) makes a rose curve. If 'a' is equal to 'b', then there is no inner loop. If 'a' is not equal to 'b', then there is a smaller loop called an 'inner loop'.
02
Graph the polar function
Graph the polar function using a graphing utility. Polar graphing involves the radial function 'r' which tells us how far from the origin to plot the point and the angle theta (\(\theta\)) which tells us the direction to that point. When graphing, notice that the 'inner loop' occurs when \(\theta\) is between \(-\pi/2\) and \(\pi/2\). Contrastingly, the 'outer loop' occurs when \(\theta\) is between \(\pi/2\) and \(-\pi/2\).
03
Find the area under the polar curve
Apply the formula for the area \(A\) under a polar curve \(r=f(\theta)\) from \(\theta=\alpha\) to \(\theta=\beta\): \(A=\frac{1}{2}\int_{\alpha}^{\beta}[f(\theta)]^2 d\theta\). Here, the integration limits for the inner loop are \(\theta=-\pi/2\) to \(\pi/2\). Hence, we will integrate \(r(\theta) = 4-6 sin\theta\) from \(-\pi/2\) to \(\pi/2\). The result will then be multiplied by \(\frac{1}{2}\).
04
Compute the definite integral
Let's compute the definite integral: \(A=\frac{1}{2}\int_{-\pi/2}^{\pi/2}(4-6 \sin \theta)^2 d\theta\). Carry out the algebra to square the integrand creating three terms, then compute the integral of each term separately. These will be simple integrals \(\int a d\theta = a\theta\) and \(\int \sin^2(\theta) d\theta\).
05
Evaluate the definite integral using Fundamental Theorem of Calculus
To evaluate the integral, subtract the antiderivative evaluated at the lower limit of integration \(-\pi/2\) from the antiderivative evaluated at the upper limit \(\pi/2\). The answer we get is the area of the region.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Polar Functions
Understanding how to graph polar functions is fundamental in mathematics, especially when considering the region they enclose. A polar function presents a relationship between the radius r, which is the distance from the origin to a point, and the angle 胃, which determines the direction from the origin to that point. Unlike Cartesian coordinates that use a grid of x and y, polar graphing is done on a circular grid.
When graphing the polar function r=4-6 sin 胃, you plot the points by calculating the radius for different angles. This will result in a 'rose curve,' and if the coefficients in the function are not equal, the graph will feature an inner loop. This loop is a smaller section of the curve, which can be graphed using technology like graphing calculators or software. For the area calculation, only the points where the curve has an inner loop are considered, which usually occurs within specific ranges of 胃.
When graphing the polar function r=4-6 sin 胃, you plot the points by calculating the radius for different angles. This will result in a 'rose curve,' and if the coefficients in the function are not equal, the graph will feature an inner loop. This loop is a smaller section of the curve, which can be graphed using technology like graphing calculators or software. For the area calculation, only the points where the curve has an inner loop are considered, which usually occurs within specific ranges of 胃.
Integral Calculus
Integral calculus is a part of mathematics that deals with finding the area under a curve, volumes, and other related quantities. It is used to sum infinitely small factors to determine a whole.
In the context of polar equations, to find the area enclosed by a polar curve, you use a specific integral calculus formula. The beauty of integral calculus lies in its ability to handle these complex shapes and curves that cannot be easily broken down into simple geometries like triangles or rectangles. The integration process adapts to the curve's nature, considering every point - hence, you can calculate precise areas of these unique shapes.
In the context of polar equations, to find the area enclosed by a polar curve, you use a specific integral calculus formula. The beauty of integral calculus lies in its ability to handle these complex shapes and curves that cannot be easily broken down into simple geometries like triangles or rectangles. The integration process adapts to the curve's nature, considering every point - hence, you can calculate precise areas of these unique shapes.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus unites the two principal concepts of calculus: differentiation and integration. It states that if a function is continuous over an interval and has an antiderivative in that interval, the definite integral of the function over that interval is equal to the difference between the values of the antiderivative at the endpoints.
This theorem is crucial when we calculate the area under the polar curve. Once the integrand is set up, we evaluate it at the limits of integration. The evaluation gives us the exact area of the shape described by the function within these limits. Without this theorem, the connection between the antiderivative and the area under the curve wouldn't be as clear, and finding areas analytically would be a much harder task.
This theorem is crucial when we calculate the area under the polar curve. Once the integrand is set up, we evaluate it at the limits of integration. The evaluation gives us the exact area of the shape described by the function within these limits. Without this theorem, the connection between the antiderivative and the area under the curve wouldn't be as clear, and finding areas analytically would be a much harder task.
Polar Coordinates
Polar coordinates are an alternative to the Cartesian coordinate system most people are familiar with. Instead of using x and y to denote a point's location, we use a radius r and an angle 胃. This makes it particularly useful for circles and shapes where symmetry about the origin plays a role.
Using polar coordinates can simplify equations for curves and shapes in the plane, especially when dealing with circular or spiral patterns. This change from Cartesian to polar often provides a more natural and simple description of a curve's properties, which is particularly evident when trying to understand the inner loops or petals of certain polar graphs.
Using polar coordinates can simplify equations for curves and shapes in the plane, especially when dealing with circular or spiral patterns. This change from Cartesian to polar often provides a more natural and simple description of a curve's properties, which is particularly evident when trying to understand the inner loops or petals of certain polar graphs.
