A Polar Double Integral Calculator is a tool that can be used to calculate double integrals for a polar function, where polar equations are used to represent a point in the polar coordinate system.
Polar Double Integrals are evaluated to find the area of the polar curve. This excellent tool solves these integrals quickly as it completely frees us from going through the complicated procedure required if solved by hand.
What Is a Polar Double Integral Calculator?
A Polar Double Integral Calculator is an online calculator that can easily solve double definite integral for any complex polar equation.
Double integration for polar point is the process of integration in which upper and lower limits for both dimensions are known. By applying double integration to the equation, we get a real definite value.
The polar equations can be algebraic or trigonometric functions of $r$ and $\theta$. Performing integration is itself a rigorous task and if one needs to evaluate a double integral over an equation, then the difficulty level of the problem increases.
Such calculations are error-prone. Therefore this friendly calculator accurately evaluates the polar integrals for you in a few seconds. It just needs the basic elements required for the calculation.
Polar systems are used in many practical fields like mathematics, engineering, and robotics, where solving these double polar integrals helps to find out the area under the polar curve.These regions are defined by the integration limits provided for each dimension. The calculator’s operation is very simple to understand. You just need a valid polar equation and integral bounds.
How To Use the Double Polar Integral Calculator?
You can use the Polar Double Integral Calculator by entering the equation, integration order, and limits in their respective areas on the calculator’s interface. Here is a detailed explanation of how to use this great tool.
Step 1
Put the polar function in the tab with the name F(R, Theta). It is a function of the two dimensions in the polar coordinate on which integration is performed.
Step 2
Select the integration order for your double integration. There are two possible orders for this type of integration. One way is to first solve concerning the radius, then concerning angle ($r dr d\theta$) or the other way around ($r d\theta dr$).
Step 3
Now enter the integral limits for radius ($r$). Put a lower limit in the R From box and an upper limit in the To box. These limits are real values of radius.
Step 4
Now input the limits for integral of angle ($\theta$). Insert lower and upper values in the Theta From and To respectively.
Step 5
Lastly, click on the Submit button. The final result shows you the mathematical representation of your problem with a finite value as the answer. This value is the measure of the area under the polar curve.
How Does the Polar Double Integral Calculator Work?
The Polar Double Integral Calculator works by collectively solving both integrals of the input function $f(r,\theta)$ under the specified intervals $r=[a,b]$ and $\theta=[c,d]$.
To understand the working of this calculator, we first need to discuss some important mathematical concepts.
What Is a Polar Coordinate System?
The Polar Coordinate system is a 2-D coordinate system where the distance of every point is determined from a fixed point. It is another pictorial representation of a point in a plane. A polar point is written as $P(r,\theta)$ and is plotted using a polar graph.
A polar point has two components. The first is the radius, which is the distance of the point from the origin, and the second is the angle, which is the direction of the point concerning the origin. So you must need these two parts to view any point in the polar system.
The polar graph is the tool to view a polar point. It is a set of concentric circles that are at an equal distance from each other representing a value of radius. The whole graph is divided into uniform sections by specified angle values.
A single point can have multiple pairs of coordinates in the polar system. Therefore, you can have the same polar interpretation for two points that are completely different from each other.The polar coordinate is a very important system for mathematical modeling. There are certain conditions in which using polar coordinates makes the calculation procedure easy and helps in better understanding.
So according to the nature of the problem, the rectangular coordinates can be converted to the polar coordinates. The formulas for the above-mentioned conversion are:
\[r = \sqrt{(x)^2 + (y)^2} \]
and
\[ \theta = tan^{-1}(\dfrac{y}{x}) \]
What Is a Double Integration?
Double integration is a kind of integration that is used for finding the regions that are constructed by two different variables. For instance, to find the region covered by the cylindrical cone in rectangular coordinates it is integrated concerning both x and y coordinates.
These coordinates have certain thresholds that describe how much the shape is expanded over the coordinate systems. Therefore, these thresholds are used in integrals.
Use of Polar Double Integrals
Polar Double Integration involves the double integration of any given function with respect to polar coordinates. When a shape is built in the polar system, it occupies some space in the coordinate system.
So to evaluate the extent of spread by the resultant polar shape, we integrate the given function over the polar variables. The unit of area in polar systems is defined as:
\[ dA = r dr d\theta \]
The formula to find the finite value of the area in the polar coordinate system is given as:
\[ Area = \int_{\theta=a}^{b} \int_{r=c}^{d} f(r,\theta) r dr d\theta \]
Solved Examples
Here are some examples solved using the polar double integral calculator.
Example 1
Take a look at the below-mentioned function:
\[ f(r,\theta) = r + 5\cos\theta \]
The order of integration for this problem is:
\[ r d\theta dr \]
The upper and lower limits for polar components are given below:
\[r = (0,1) \]
and
\[ \theta = (0,2\pi) \]
Solution
Use our calculator to solve the integrals as:
\[ \int_{r=0}^{1} \int_{\theta=0}^{2\pi} r + 5\cos\theta r d\theta dr = 2\pi = 6.28319 \]
Example 2
Consider the following function:
\[ f(r,\theta) = r^2\sin\theta \]
The order of integration for this problem is:
\[ r dr d\theta \]
The limits for polar variables are as follows:
\[r = 0,1+\cos\theta \]
and
\[ \theta = (0,\pi) \]
Solution
Our calculator gives the answer in fraction and its equivalent decimal number:
\[ \int_{\theta=0}^{\pi} \int_{r=0}^{1+\cos\theta} r^2\sin\theta r dr d\theta = \dfrac{8}{5} = 1.6 \]
