Convert Double Integral to Polar Coordinates Calculator + Online Solver With Free Steps (2025)

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 equationssince 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.

Convert Double Integral to Polar Coordinates Calculator + Online Solver With Free Steps (1)

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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Convert Double Integral to Polar Coordinates Calculator + Online Solver With Free Steps (2025)

FAQs

How to solve a double integral by converting to polar coordinates? ›

The double integral ∬Df(x,y)dA in rectangular coordinates can be converted to a double integral in polar coordinates as ∬Df(rcos(θ),rsin(θ))rdrdθ.

What is the best calculator for double integrals? ›

Wolfram|Alpha is a great tool for calculating indefinite and definite double integrals.

What is the symbol for a double integral? ›

Double integral is mainly used to find the surface area of a 2d figure, and it is denoted using ' ∫∫'. We can easily find the area of a rectangular region by double integration. If we know simple integration, then it will be easy to solve double integration problems.

How do you find the volume of a double integral? ›

The volume V is V=∬D(f(x,y)−g(x,y))dxdy=∬D(3−3x2−3y2)dxdy. This integral is very simple to calculate if you know how to change variables to polar coordinates. (If you don't yet know how to do this, you can still calculate the integral if you are good at doing integrals, but it gets pretty ugly.

How do you convert an integral to a polar coordinate? ›

Use x=rcosθ,y=rsinθ, and dA=rdrdθ to convert an integral in rectangular coordinates to an integral in polar coordinates. Use r2=x2+y2 and θ=tan−1(yx) to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.

How to convert into polar coordinates? ›

To convert from polar co-ordinates to Cartesian co-ordinates, use the equations x = r cos θ , y = r sin θ . To convert from Cartesian co-ordinates to polar co-ordinates, use the equations r2 = x2 + y2 , tan θ = y x .

What is the most accurate integral calculator? ›

Symbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more.

What scientific calculator can solve integrals? ›

Wolfram|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition. Learn more about: Integrals.

Can desmos do double integrals? ›

In Desmos 3D you can take double integrals. For example, try setting f(x,y)=xy2 f ( x , y ) = x y 2 and taking the double integral of f(x,y)dxdy f ( x , y ) d x d y where the bounds for x are from 0 to 2 and the bounds for yarefrom\(0 y a r e f r o m \( 0 to 1 .

What is the formula for a double integral? ›

Double integral is a way to integrate over a two-dimensional area. Double Integral containing two variables over a region R=[a, b]×[c, d] can be defined as, ∫Rf(x, y)dA=∫ba∫dcf(x, y) dy dx. In simple words here we first solve the inner integral and then the outer integral, thus the name Double Integral.

What does R mean in a double integral? ›

Instead we have the R written below the two integrals to denote the region that we are integrating over. As indicated above one interpretation of the double integral of f(x,y) f ( x , y ) over the rectangle R is the volume under the function f(x,y) f ( x , y ) (and above the xy-plane).

What is the law of double integral? ›

A double integral is an integral of a two-variable function f (x, y) over a region R. If R = [a, b] × [c, d], then the double integral can be done by iterated integration (integrate first with respect to y, and then integrate with respect to x).

What is double integral calculator? ›

Double Integral Calculator is a free online tool that displays the value for the double integral function. BYJU'S online double integral calculator tool makes the calculation faster, and it displays the double integral value in a fraction of seconds.

How do you calculate double integrals? ›

Double integrals measure volume, and are defined as limits of double Riemann Sums. We can estimate them by forgetting about the limit, and just looking at a Riemann sum; essentially this means we're adding up the volume of boxes that fit "under" the surface z=f(x,y).

How to find the bounds of a double integral in polar coordinates? ›

You get the bounds simpl by putting the value or taking the limit for each coordinate. For example say r=√x2+y2 and say that x,y∈[0,2]. Then you look at the maximum and minimum of r as a function of x and y. The solution will be an interval because r(x,y) is continuous.

What is the double integral of a polar region? ›

In the case of double integral in polar coordinates we made the connection dA=dxdy. dxdy is the area of an infinitesimal rectangle between x and x+dx and y and y+dy. In polar coordinates, dA=rd(theta)dr is the area of an infinitesimal sector between r and r+dr and theta and theta+d(theta).

How do you convert DX to polar coordinates? ›

To change the function and limits of integration from rectangular coordinates to polar coordinates, we'll use the conversion formulas x=rcos(theta), y=rsin(theta), and r^2=x^2+y^2. Remember also that when you convert dA or dy dx to polar coordinates, it converts as dA=dy dx=r dr dtheta.

What is the formula for integral in polar coordinates? ›

Integrate from the lowest value of θ for which the corresponding ray intersects R to the highest value of θ. x + y = 1 → r cos θ + r sin θ = 1, or r = 1 cos θ + sin θ .

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