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Convert Double Integral to Polar Coordinates Calculator + Online Solver With Free Steps


The Double Integral to Polar Coordinates Calculator calculates the double integral of a given equation with the integral given as $\int_{\theta1}^{\theta2}\int_{r1}^{r2} f(r, \theta)\, r\, dr\, d\theta$. 

Furthermore, this calculator can calculate any legitimate equation for which the double integral works. The limits of the integrals are as per the requirements in the original equation. 

Moreover, if the limits are continuous, the calculator will still calculate the double integral for polar-continuous coordinates and give the correct result. The order of the integral is changeable, where you can first integrate the “r” variable or the angle variable. 

The calculator can also calculate simple linear line equations since they are first converted into the polar form. These equations also require the user to find the angle 𝛉 before using the calculator to utilize its smooth operation.

What Is the Double Integral to Polar Coordinates Calculator?

The Double Integral to Polar Coordinates Calculator is an online tool that utilizes the double integration of a polar equation by integrating the equation with the angular and the radial variable. Additionally, the integration result is expressed in a fractional form and later converted to an approximate decimal number. 

The calculator consists of 5 single-line text boxes and one dropdown menu. The first text box, labeled as “f(r,q) (q in radians),” is where you can enter the polar equation for the area you want to find using the double integral.

Furthermore, the limits for the radius value “r” is entered in the two text boxes labeled as given, and the last two text boxes are for entering the range of the angles. The dropdown menu, labeled as “order,” consists of two orders of the integration to select whether to integrate the angle variable, “q,” or the radius variable, “r,” first.  

How To Use the Double Integral to Polar Coordinates  Calculator?

You can use the Double Integral to Polar Coordinates Calculator by simply entering the equation into the textbox and setting up the limits for the radius and angles of the integral you want to find. The order does not matter as much as both orders usually result in the same answer and the end. Then you press the submit button to acquire the result. 

The stepwise guidelines for the calculator’s usage are mentioned below.

Step 1

Enter the Desired equation for which you want to find the polar integral and ensure its legitimacy.

Step 2

After that, enter the limits for both the variables, “r” and “q, to set the integration range. Furthermore, ensure that the limits for the angle variable are in radians and not degrees. 

Step 3

Press the “Submit” button to get the results. 

Results

A pop-up window appears after pressing the submit button. This window consists of a single section labeled “Definite Integral.” This section first depicts the integral equation, as entered by the user, with the double integrals and the limits. This way, the user can verify the correctness of his input in the text boxes. 

Furthermore, the result is calculated and expressed in fractional form. Later on, it is approximated into the decimal form for better readability. You can use the “More Digits” button on the top right to add more decimal places to the decimal approximation. 

Solved Examples

Example 1

A circle of an equation $ x^2 + y^2 = 9$ is given on the cartesian plane. Find the area of the section that is covered by 1 < r ≤ 3, and the angle range is given as 0 < 𝛉 ≤ π.

Solution

First of all, we convert this Cartesian equation that is $ x^2 + y^2 = 9$ to a Polar equation 

As we know, x$^2$ + y$^2$ = r$^2$

Hence, 

\[ f(r, \theta) = 9 – r^2 \]

Using this equation in the double integral calculation given as:

 \[\int_{\theta1}^{\theta2}\int_{r1}^{r2} f(r, \theta)\, r\, dr\, d\theta\] 

\[\int_{0}^{\pi}\int_{1}^{3} (9 – r^2) \, r\, dr\, d\theta\] 

\[\int_{0}^{\pi} -\frac{(r^2 – 9)^2}{4}\bigg\vert_1^3 \, d\theta\] 

\[\int_{0}^{\pi} -\frac{(3^2 – 9)^2}{4} – \left(-\frac{(1^2 – 9)^2}{4}\right) \, d\theta\]

\[\int_{0}^{\pi} 16 \, d\theta\]

\[ 16\theta\bigg\vert_0^\pi \, \]

\[ 16 (\pi – 0)\]

\[ \mathbf{16 \pi }\]

Thus, the Area covered by the radius 1 and 3 with the angles from 0 to 𝝅 is equal to 16𝝅. This can be further approximated into a decimal form that is given as 50.265.

Example 2

Consider an equation of $x^2 + y^2 =  16 $. Find its area for the part enclosed in 1 < r ≤ cos(2π) and -π/2 < 𝛉 ≤ π/2.

Solution

First of all, we convert this Cartesian equation that is $ x^2 + y^2 = 16$ to a Polar equation 

We here have a limit as cos (2π) which can also be calculated similarly to the last example

As we know, $ x^2 + y^2 = r^2$

Hence, 

\[ f(r, \theta) = 16 – r^2 \]

Using this equation in the double integral calculation given as:

 \[\int_{\theta1}^{\theta2}\int_{r1}^{r2} f(r, \theta)\, r\, dr\, d\theta\] 

\[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{\cos{2\pi}} (16 – r^2) \, r\, dr\, d\theta\] 

\[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} -\frac{(r^2 – 16)^2}{4}\bigg\vert_0^{\cos{2\pi}} \, d\theta\] 

\[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} -\frac{((\cos{2\pi})^2 – 16)^2}{4} – \left(-\frac{(0^2 – 16)^2}{4}\right) \, d\theta\]

\[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{31}{4} \, d\theta\]

\[ \frac{31}{4}\theta\bigg\vert_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \, \]

\[ \frac{31}{4} \left(\frac{\pi}{2} – \left(-\frac{\pi}{2}\right)\right)\]

\[ \mathbf{\frac{31 \pi}{4} }\]

Thus, the Area covered by the radius 1 and 3 with the angles from 0 to is equal to 31𝝅/4. This can be further approximated into a decimal form given as 24.347.

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