# Integral calculus: plane areas in polar coordinates

• delapcsoncruz
Pythagorean theoremIn summary, when finding the area inside the graph of r=2sinθ and outside the graph of r=sinθ+cosθ, the person encountered difficulty in computing the values of 'r' and finding the intersection points. They were advised to use the useful tip of multiplying both sides by 'r' and converting to Cartesian coordinates, which led to the discovery of two additional intersection points. The person was also given an example of using this tip with r=2sinθ and instructed to convert to Cartesian coordinates using the Pythagorean theorem. This resulted in finding the area inside the graph.
delapcsoncruz
what is the area inside the graph of r=2sinθ and outside the graph of r=sinθ+cosθ?

so i compute for the values of 'r',... but, i only got one intersection point which is (45°, 1.41).
there must be two intersection points right? but I've only got one. what shall i do?
i cannot compute for the area of the said region because I've only got one limit which is ∏/4.. what shall i do?

thanks a lot..

hi delapcsoncruz!

useful tip: multiply both sides by r, then convert to cartesian coordinates …

that (amost) immediately gives you the positions of these circles

Alternatively not that 5pi/4 is also an intersection point

Office_Shredder said:
Alternatively not that 5pi/4 is also an intersection point

the value of 'r' in 5pi/4 is -1.41 , so that is also equal to pi/4 which r is 1.41

try it with r = 2sinθ …

what do you get?

r^2=2rsin(theta)

now convert to cartesian (x and y)

## 1. What is integral calculus?

Integral calculus is a branch of mathematics that deals with calculating and analyzing the area under a curve. It involves finding the antiderivative of a function, which represents the area under the curve. It is used to solve problems related to motion, area, volume, and other physical quantities.

## 2. What are plane areas in polar coordinates?

Plane areas in polar coordinates refer to the area enclosed by a curve in the polar coordinate system. In this system, points are represented by a distance from the origin and an angle from a fixed reference line. The area is calculated by integrating the function in polar form over a specific interval.

## 3. How is integral calculus used to find plane areas in polar coordinates?

To find the area enclosed by a curve in polar coordinates, we need to convert the function into polar form and then integrate it over a specific interval. This integration gives us the area under the curve, which represents the plane area in polar coordinates.

## 4. What are the applications of integral calculus in finding plane areas in polar coordinates?

The applications of integral calculus in finding plane areas in polar coordinates are numerous. It is used in physics and engineering to calculate the moments of inertia, center of mass, and work done by a force. It is also used in economics, biology, and other fields to model and analyze various phenomena.

## 5. Are there any limitations to using integral calculus for finding plane areas in polar coordinates?

Yes, there are certain limitations when using integral calculus for finding plane areas in polar coordinates. The function must be continuous and defined over the given interval, and the curve should not intersect itself. Also, the integration process can be complex for certain functions, making it difficult to find the exact area. In such cases, numerical methods are used to approximate the area under the curve.

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