Find what you need to study
3 min read•january 25, 2023
Sumi Vora
Jed Quiaoit
Sumi Vora
Jed Quiaoit
Embarking on the adventure of AP Calculus, we often encounter the mesmerizing world of shapes, curves, and areas that seem to dance between dimensions. Among these, the concept of polar coordinates offers a fresh perspective on understanding the geometry of curves.
Our mission is to master the technique of calculating areas of regions defined by polar curves, using definite integrals. By the end of this guide, you'll be able to wrap your head around polar curves and the areas they enclose with confidence and curiosity.
Before we dive into calculating areas, let's go over what polar coordinates are. Unlike the familiar Cartesian coordinates (x and y), which locate points through horizontal and vertical distances, polar coordinates use a radius (r) and an angle (θ) to pinpoint the location of a point in a plane. This system is incredibly useful for describing curves that are circles or spirals, which are difficult to express in Cartesian terms.
To understand the area under a polar curve, we must first grasp how to express the concept of area in polar terms. The area of a sector (a pizza slice of a circle) is a fundamental building block. In polar coordinates, the area of a sector with radius and angle (in radians) is given by .
The beauty of calculus shines when we apply the concept of integration to polar coordinates. To find the area enclosed by a polar curve from to , we use the definite integral:
This formula is a direct extension of finding the area of a sector, but instead of a single slice, we sum up infinitely small slices (sectors) between and , each with its own radius determined by the polar function .
Problem Statement
Find the area inside the circle over the range .
Here are the steps we must follow:
So, the area inside the curve over the range , is square units.
Calculate the area of one petal of the rose curve .
Here are the steps we must follow:
So, the area of one petal of the rose curve is units squared.
Let’s try a practice problem to test your new skills!
Calculate the area enclosed by the polar curve over the interval .
Sketch the Curve
Always start by sketching the curve to understand its shape and symmetry. For , it’s a limacon with an inner loop.
Inner Symmetry
Note that the curve is symmetrical about the horizontal axis; this, you can calculate the area for half the curve and then double it.
Set Up the Integral for Half the Curve
Since you’re calculating the area for half the curve (from to ) and then doubling it, set up the integral as follows :
Simplify and Solve the Integral
Simplify the integral expression first:
Then, solve each term of the integral separately:
Combine
Combine all terms to find the total area:
So, the area enclosed by the polar curve is is units squared.
Understanding how to find the area of a polar region or the area bounded by a single polar curve expands our problem-solving toolkit in calculus. It allows us to tackle a variety of problems involving curves in polar form with precision and accuracy.
Area under the curve
: The area under the curve refers to the total amount of space enclosed between a function and the x-axis on a given interval.cosθ
: The cosine of an angle θ is the ratio of the length of the adjacent side to θ in a right triangle to the length of the hypotenuse.Enclosed area between polar curves
: The enclosed area between two polar curves is the region in the plane that is bounded by these curves. It represents the area of overlap between the two curves.Integration
: Integration is an operation in calculus that finds antiderivatives, or reverse derivatives, allowing us to calculate areas, volumes, and other quantities related to curves and functions.Polar Coordinates
: Polar coordinates are a two-dimensional coordinate system used to locate points in space using radial distance (r) and angular displacement (θ) from a reference point called the pole.Polar Functions
: Polar functions are mathematical equations that describe curves in terms of a distance from the origin and an angle. They are often used to represent shapes that have radial symmetry.r^n
: In mathematics, r^n refers to raising a number r (the base) to an exponent n. It means multiplying r by itself n times.sinθ
: The sine of an angle θ is the ratio of the length of the side opposite to θ in a right triangle to the length of the hypotenuse.3 min read•january 25, 2023
Sumi Vora
Jed Quiaoit
Sumi Vora
Jed Quiaoit
Embarking on the adventure of AP Calculus, we often encounter the mesmerizing world of shapes, curves, and areas that seem to dance between dimensions. Among these, the concept of polar coordinates offers a fresh perspective on understanding the geometry of curves.
Our mission is to master the technique of calculating areas of regions defined by polar curves, using definite integrals. By the end of this guide, you'll be able to wrap your head around polar curves and the areas they enclose with confidence and curiosity.
Before we dive into calculating areas, let's go over what polar coordinates are. Unlike the familiar Cartesian coordinates (x and y), which locate points through horizontal and vertical distances, polar coordinates use a radius (r) and an angle (θ) to pinpoint the location of a point in a plane. This system is incredibly useful for describing curves that are circles or spirals, which are difficult to express in Cartesian terms.
To understand the area under a polar curve, we must first grasp how to express the concept of area in polar terms. The area of a sector (a pizza slice of a circle) is a fundamental building block. In polar coordinates, the area of a sector with radius and angle (in radians) is given by .
The beauty of calculus shines when we apply the concept of integration to polar coordinates. To find the area enclosed by a polar curve from to , we use the definite integral:
This formula is a direct extension of finding the area of a sector, but instead of a single slice, we sum up infinitely small slices (sectors) between and , each with its own radius determined by the polar function .
Problem Statement
Find the area inside the circle over the range .
Here are the steps we must follow:
So, the area inside the curve over the range , is square units.
Calculate the area of one petal of the rose curve .
Here are the steps we must follow:
So, the area of one petal of the rose curve is units squared.
Let’s try a practice problem to test your new skills!
Calculate the area enclosed by the polar curve over the interval .
Sketch the Curve
Always start by sketching the curve to understand its shape and symmetry. For , it’s a limacon with an inner loop.
Inner Symmetry
Note that the curve is symmetrical about the horizontal axis; this, you can calculate the area for half the curve and then double it.
Set Up the Integral for Half the Curve
Since you’re calculating the area for half the curve (from to ) and then doubling it, set up the integral as follows :
Simplify and Solve the Integral
Simplify the integral expression first:
Then, solve each term of the integral separately:
Combine
Combine all terms to find the total area:
So, the area enclosed by the polar curve is is units squared.
Understanding how to find the area of a polar region or the area bounded by a single polar curve expands our problem-solving toolkit in calculus. It allows us to tackle a variety of problems involving curves in polar form with precision and accuracy.
Area under the curve
: The area under the curve refers to the total amount of space enclosed between a function and the x-axis on a given interval.cosθ
: The cosine of an angle θ is the ratio of the length of the adjacent side to θ in a right triangle to the length of the hypotenuse.Enclosed area between polar curves
: The enclosed area between two polar curves is the region in the plane that is bounded by these curves. It represents the area of overlap between the two curves.Integration
: Integration is an operation in calculus that finds antiderivatives, or reverse derivatives, allowing us to calculate areas, volumes, and other quantities related to curves and functions.Polar Coordinates
: Polar coordinates are a two-dimensional coordinate system used to locate points in space using radial distance (r) and angular displacement (θ) from a reference point called the pole.Polar Functions
: Polar functions are mathematical equations that describe curves in terms of a distance from the origin and an angle. They are often used to represent shapes that have radial symmetry.r^n
: In mathematics, r^n refers to raising a number r (the base) to an exponent n. It means multiplying r by itself n times.sinθ
: The sine of an angle θ is the ratio of the length of the side opposite to θ in a right triangle to the length of the hypotenuse.© 2024 Fiveable Inc. All rights reserved.
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