# Double Integral Calculator: Definition, Examples

A double integral calculator is a web-based app. It allows you to assess integrals online. It is also famous as a multiple integral calculator since it evaluates many integrals. So, using online integration solvers may be quite beneficial since they give quick results with steps and graphs, among other things.

The step-by-step double integral calculator computes both multiple definite and multiple indefinite integrals. Because a simple integration calculator with steps cannot evaluate several integers, this calculator comes in handy. In this article, we are talking about this calculator. So, please keep reading to know more about it.

## Double Integral Calculator definition

The Double Integral Calculator is used to compute the value of a double integral. Double integrals can be used to calculate the area of a two-dimensional figure. The symbol for double integration is ‘∫∫‘.

Moreover, the Double Integral Calculator is an online tool for integrating a given function and calculating the value of the double integral. Then, double integrals can be used to calculate the volume under a surface and the average value of a two-variable function. Then, enter the data in the input fields to use the Double Integral Calculator.

## Double Integral Calculator with steps

Please use the online double integral calculator to get the value of the double integral by following the procedures outlined below:

Step 1: Open any Double Integral Calculator online.

Step 2: Fill in the function and the limitations in the provided input boxes. Choose which variable will be integrated first from the drop-down selection.

Step 3: To obtain the value of the double integral, click the “Calculate” button.

Step 4: Select “Reset” to clear the fields and input new values.

## Double Integral Calculator with solution

Integral calculus is made up of many forms of integrations, including basic integration, double integration, and triple integration. When dealing with a function in a single variable, the integration is performed over an interval (one-dimensional space). When we have a function that is dependent on two variables, we basically integrate it across an area (2-dimensional space).

If we have a double integral expressed as ∫ d c ∫ b a f ( x , y ) d x d y, we may get its value using the procedures below.

• First, we begin by calculating the inner integral. Because dx precedes dy, we shall first integrate the function with regard to x. Then, all terms containing the letter y will be considered as constants.
• Then, the definite integral’s inner bounds are used. Our function will now only be defined in terms of y.
• Also, the outer integral is then solved. Then, this means we’re integrating the function in relation to y.
• Then, to obtain the final result, apply the outer integral’s boundaries.

As a side remark, the integral value of ∫ d c ∫ b a f ( x , y ) d x d y is equivalent to ∫ b a ∫ d c f ( x , y ) d y d x.

## Double Integral Calculator examples

Example 1: Determine the double integral value of I =  ∫ 1 0 ∫ 3 2 x 3 dxdy and then double-check it using the double integral calculator.

Solution:

I =  ∫ 1 0 ∫ 3 2 x 3 y dxdy

We begin by integrating the function in relation to x.

I =  ∫ 1 0 [ ∫ 3 2 x 3 y d x ] d y

I =  ∫ 1 0 [ x 4 y 4 ] 3 2 d y

We now integrate the function with regard to y.

I =   ∫ 1 0 65 y 4 d y

I =   [ 65 y 2 8 ] 1 0

Then, I = 65/8

I = 8.125

Example 2: Calculate the double integral value of ∫ 8.2 6 ∫ 2 1 ( x y − y ) d y d x  and then double-check it using the double integral calculator.

Solution:

∫ 8.2 6 ∫ 2 1 ( x y − y ) d y d x

We begin by integrating the function in relation to y.

I =  ∫ 8.2 6 [ y 2 x 2 − y 2 2 ] 2 1 d x

We now integrate the function with regard to x.

I =  ∫ 8.2 6 [ 3 x 2 − 3 2 ] d x

Then, I =   [ 3 x 2 4 − 3 x 2 ] 8.2 6

Atlast, I = 20.13

Similarly, you can use the double integral calculator to calculate the values of the following double integrals:

∫ 5.5 3.2 ∫ 7 6 x 2 y 3 d y d x

So, ∫ 5 2 ∫ 13 8 [ x 2 y + x y 2 ] d x d y

## Double Integral calculator polar

• Recognize the format of a polar rectangular region’s double integral.
• Using an iterated integral, evaluate a double integral in polar coordinates.
• Recognize the format of a double integral across a broad polar range.
• Calculate areas and volumes using double integrals in polar coordinates.

When we switch from rectangular to polar coordinates, double integrals become considerably easier to analyse. However, before we go into detail about how to make this adjustment, we must first define the idea of a double integral in a polar rectangular region.

The double integration in polar coordinates is:

∫^{θ_2} _{θ_1} ∫^{r_2} _{r_1} f (r, θ) dθ, dr

To begin, we must integrate the f(theta, r) with respect to r within the limits (r 1 and r 2), where theta is constant, then integrate the resultant equation as from (theta_1 to theta _2, text where r 1 and r 2 ) are constant.

We split R into subrectangles with sides parallel to the coordinate axes while defining the double integral for a continuous function in rectangular coordinates—say, g over a region R in the xy -plane. These sides all have constant x and/or constant y values. The shape we deal with in polar coordinates is a polar rectangle with constant r -values and/or constant -values on its sides.

### Example 1

Draw a polar rectangle area.

R={(r,θ)|1≤r≤3,0≤θ≤π}.

Solution:

Let, r=1 and r=3 are circles of radius 1 and 3, respectively, and 0 spans the whole upper half of the plane. As a result, the area R resembles a semicircular band.

With inner radius 1 and outer radius 3, half an annulus R is drawn. That is, x squared + y squared = 1 gives the inner semicircle, but x squared + y squared = 9 gives the outer semicircle. The polar area R is bounded by two semicircles.

Now that we’ve drawn a polar rectangular rectangle, let’s show how to use polar coordinates to calculate a double integral over this region.

### Example 2

Calculate the integral.

∬R(1−x2−y2)dA, where R is the xy-plane unit circle.

Solution:

Because the area R is a unit circle, we may write it as {(r,θ)|0≤r≤1,0≤θ≤2π}.

We get dA=rdrd by converting x=rcosθ,y=rsinθ and dA=rd d theta.

∬R(1−x2−y2)dA=∫2π0∫10(1−r2)rdrdθ

=∫2π0∫10(r−r3)drdθ

Then, =∫2π0[r22−r44]10dθ

=∫2π014dθ=π2.

## Double integral calculator with limits

### Example 1

Calculate the integral ∬Dxy^2dA,  where D is the rectangle defined by 0≤x≤2 and 0≤y≤1, as shown below.

Solution: The double integral will be computed as the iterated integral ∫ 1 0 (∫ 2 0 xy^2dx)dy.

Within the parenthesis. So, we first integrate with respect to x. Then, during this integration stage, we must regard y as a constant, as we did with partial derivatives. Because the integral of cx for any constant c is cx^2/2, we derive

∫ 1 0 (∫ 2 0 xy^2dx)dy=∫ 1 0 (x^2/2 y^2∣∣∣x=2x x=0) dy

Then, =∫ 1 0 (2^2/2 y^2−0^2/2 y^2) dy

Then, =∫ 1 0 2y^2dy.

So, it’s worth noting that in the first line above, we put the limits as x=2 and x=0 to make it clear that x is the variable we just integrated.

Finally, we must compute the integral with respect to y, which is straightforward. Since x is no longer present, it is simply a one-variable integral. Our double integral is

∬Dxy2dA=∫102y2dy

Then, =2y^3/3∣∣∣10=2(1^3)3−2(0^3)3=2/3.

Moreover, to double-check our solution, we may compute the integral in the other way. At first with respect to y and then with respect to x. So, the only secret is to keep in mind that while integrating with regard to y, we must consider x to be a constant. Because the integral of cy^2 for any constant c is cy^3/3, we derive

∬Dxy^2dA=∫20(∫10xy2dy)dx

Then, =∫ 2 0 (xy^3/3∣∣∣y=1y=0)dx

Then, =∫2 0 x/3 dx=x^2/6∣∣∣2 0=4/6=2/3.

So, as it should, this iterated integral yields the same result.

### Example 2

Moreover, rectangular areas are simple because the bounds  (a≤x≤b and c≤y≤d) are fixed, implying that the x and y ranges do not rely on each other. Then, for different forms of regions, the range of one variable will be dependent on the range of the other. Then, here is an example of integrating across the region described by 0≤x≤2 and 0≤y≤x/2. Because the range of y depends on x, this region is not a rectangle. So, the territory is, in reality, the triangle depicted below.

Moreover, integrating x first, a double integral is computed across a triangular area.

Calculate ∬DfdA  using the same function f(x,y)=xy^2 as in Example 1, where D is the above triangle.

Solution:

Because the bounds of one variable rely on the other, a triangle is slightly more difficult than a rectangle. Then, the bounds of y depend on x for the triangle formed by 0≤x≤2 and 0≤y≤x/2. So, as seen above by the vertical dashed line from (x,0) to (x,x/2) for a given value of x, y ranges from 0 to x/2.

Moreover, the outer bounds of a double integral must be constant, whereas the inner limits can vary depending on the outer variable. Then, this indicates that, like in the second method, we must use y as the inner integration variables. Then, the sole difference between this example and Example 1 is that the top limit of y is x/2. So, the double integral is

∬Dxy2dA=∫20(∫x/20xy2dy)dx

=∫ 2 0 (x^3y^3∣∣y=x/2y=0)dx

Then, =∫ 2 0 (x3(x2)3−x303)dx

Then, =∫ 2 0x424dx

Atlast, =x^5/5⋅24∣∣∣2 0 =32/5⋅24=4/15.

## Double integral calculator with graph

Moreover, in this section, we will go over some of the most significant formulas and principles that are employed by double integral calculators to execute double integration. Then, you must examine several strategies for addressing integration issues, including integration by replacements and integration by parts or formulae. Then, the rule for double integration by parts in double integrals is: ∫∫m dn/dx dx . dy = ∫[mn -∫n dm/dx dx] dy

Here are some of the most essential features of the double integral:

∫_ {x = a} ^b ∫_ {y = c} ^d f (x, y) dy . dx = ∫_ {y = c} ^d ∫_ {x = a} ^b f (x, y) dx . dy

∫∫( f(x, y) ± g (x, y)) dA = ∫∫ g (x,y) ± dA ∫∫ f(x, y) dA

If f(x, y) less than g(x, y), then ∫∫g (x, y) dA greater than ∫∫ f(x, y) dA

k ∫∫ f (x, y) . dA = ∫∫ k. f(x ,y). dA

∫∫ R ∪ S f (x, y) . dA = ∫∫ R f (x, y). dA + ∫∫ sf (x, y). dA

Moreover, when we need to discover the double integration of variable M, we define M = f(x, y) across the domain K in the xy plane. So, if we identify the endpoints for x and y as the region’s bounds and split the region into vertical stripes, we may apply the formula:

∫∫_K f(x, y) dA = ∫^{x = b} _{x = a} ∫^{y = f_2 (x)} _{y = f_1 (x)} f(x, y) dy dx

### Function

If the function m is the continuous function, then the following is true:

∫∫_K f(x, y) dA = ∫^{x = b} _{x = a} ∫^{y = f_2 (x)} _{y = f_1 (x)} f(x, y) dy dx = ∫^{x = d} _{x = c} ∫^{x = n_2 (y)} _{x = n_1 (y)} f(x, y) dx dy

An online Integral Calculator, on the other hand, allows you to assess the integrals of the specified functions with respect to the variable in question.

## Double Integral calculator without limits

### Example 1

Find the value of ∫ (9x+ 25)/ (x+3)2 .dx

Solution:

(9x+ 25)/ (x+3)2 is a rational function.

We get (9x+ 25)/(x+3)2 = A/(x+3) + B/(x+3)2 using the partial fraction decomposition.

Taking LCD, we get

(9x+ 25)/(x+3)2 = [A(x+3) +B]/(x+3)2

When we multiply the numerator by the denominator, we obtain

9x+ 25 = A(x+3) +B

Solving for B when x = -3, we get B = -2

Solving for A when x = 0, we get A = 9

As a result, the partial fraction may be decomposed as 9/(x+3) -2/(x+3)^2

Find the integral of 9/(x+3) -2/(x+3)2 as indicated in the integration formulae above.

∫[9/(x+3)]

dx – ∫ -2(x+3)^2 .dx

= 9 ln(x+3) – 2 /(x+3) +C

### Example 2

Question 2:- Solve the function ∫∫x.log x.dx.dy

Solution: Assume I = ∫∫x.log x.dx.dy

To begin, compute the inner integral of the functions ∫x.log x dx.

We can solve the given integral using the rule of integration by parts.;

∫x.log x dx = ∫(log x)x dx

The logarithmic function on the x function is not immediately integrated with another function x. As a result, we can interpret it as follows:

u = log x and dv = x dx

So, du = (1/x) dx.v = x2/2

Therefore, ∫x.log x.dx = (log x)x2/2 − ∫(x2/2)(1/x)dx

= x2/2(log x)−1/2 ∫x.dx

Hence, ∫x.log x.dx = x2/2(log x) − 1/4×2

This is the outcome of integrating by parts for the inner integral. Now, use it with the outer integral functions as follows:

∫∫x.log x.dxdy = ∫[x2/2(log x) − 1/4×2]dy

I = ∫[x2/2(log x)] dy − ∫[1/4×2] dy

I = (x2y/2) (log x) – 1/4x2y.dy + C

### How do you solve double integrals on a calculator?

• Atfirst, open any Double Integral Calculator online.
• Then, fill in the function and the limitations in the provided input boxes. Choose which variable will be integrated first from the drop-down selection.
• Then, to obtain the value of the double integral, click the “Calculate” button.
• Atlast, select “Reset” to clear the fields and input new values.

### How do you integrate a double integral?

The outer bounds of a double integral must be constant, whereas the inner limits can vary depending on the outer variable. This indicates that, like in the second method of computing Example 1, we must use y as the inner integration variables.

### What is the physical interpretation of double integral?

An Integral is a function that computes the area between two curves. A double integral calculates the volume under the surface taken between two points. One surface is used as a reference, and the volume of the surface under examination is calculated by integrating it.

### What is a triple integral calculator?

Moreover, the Triple Integral Calculator is a free online tool that displays the integrated value of a function. Also, any online triple integral calculator tool will speed up the calculation and display the integrated number in a matter of seconds.

### What is the difference between iterated and double integral?

When we are given a double integral, we want to convert it to an iterated integral so that we can simply evaluate one integral at a time, as we would in single variable calculus. We always assess iterated integrals from the inside out.

### What is Interval Notation Calculator?

However, the Interval Notation Calculator is a free online tool. However, it displays the number line for a given interval. Also, the online interval notation calculator tool accelerates the calculation. Also, it displays the number line and inequality form in seconds. Moreover, the Interval Notation Calculator computes the length of any interval between two integers. Then, it represents the inequality based on the topology chosen. Moreover, Interval notation is a method of expressing real number line subsets.

### What does a double integral represent?

The volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z=f(x,y)) z = f ( x , y ) ) and the plane containing its domain is represented by the double integral of a positive function of two variables.

### What are the integration rules?

 Common Functions Function Integral Constant ∫a dx ax + C Variable ∫x dx x2/2 + C Square ∫x2 dx x3/3 + C Reciprocal ∫(1/x) dx ln|x| + C Exponential ∫ex dx ex + C ∫ax dx ax/ln(a) + C ∫ln(x) dx x ln(x) − x + C Trigonometry (x in radians) ∫cos(x) dx sin(x) + C ∫sin(x) dx -cos(x) + C ∫sec2(x) dx tan(x) + C

### What is meant by an iterated integral?

An integral of a function with many variables that is evaluated by determining the definite integral with respect to one variable, then the definite integral of the result with respect to the second, and so on until the requisite precision is obtained.

### What is a double integral calculator?

Moreover, the Double Integral Calculator is an online tool for integrating a given function and calculating the value of the double integral. Then, double integrals can be used to calculate the volume under a surface and the average value of a two-variable function. Then, enter the data in the input fields to use the Double Integral Calculator.