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# Double Integral Calculator

To use double integral calculator, select the type of integral, enter values into the required input fields, and click calculate button

## Double integral calculator

Double integral calculator is used to find the integral of a double variable function. This calculator takes the 2-dimensional function and provides the step-by-step solution with respect to both variables. This double integration calculator will solve the double definite and indefinite problems easily.

## What is double integral?

In calculus, the double integral is a technique or method for finding the integral of two variable functions in 2-dimension. It is used to evaluate the volume and area over the region in R^{2}. The double variable function can be written as f(x, y) and is denoted in form of an integral as:

**∫∫ _{R} f(x, y) dx dy**

The limit values should be applied in the case of definite integrals while in indefinite integrals the boundary values are not used.

## How to evaluate the double integral problems?

The double integral calculator above is a helpful way to evaluate double integral problems. But if you want to evaluate them manually let us take an example.

**Example**

Evaluate the double integral of the given function.

f(x, y) = 3x^{2}y + 2y

**Solution**

**Step 1: **Apply the double integral notation to the given function.

∫∫ f(x, y) dxdy = ∫∫ [3x^{2}y + 2y] dxdy

**Step 2: **Integrate the above expression with respect to “x”

∫∫ [3x^{2}y + 2y] dxdy = ∫ [ ∫[3x^{2}y + 2y] dx] dy …. (1)

**For “x”**

∫[3x^{2}y + 2y] dx = ∫[3x^{2}y] dx + ∫[2y] dx

∫[3x^{2}y + 2y] dx = 3y∫[x^{2}] dx + 2y∫[1] dx

∫[3x^{2}y + 2y] dx = 3y [x^{2+1}/2+1] + 2y[x]

∫[3x^{2}y + 2y] dx = 3y [x^{3}/3] + 2y[x]

∫[3x^{2}y + 2y] dx = 3x^{3}y/3 + 2xy

∫[3x^{2}y + 2y] dx = x^{3}y + 2xy

**Step 3: **Put the integral of “x” in 1 and integrate the expression for “y”.

∫∫ [3x^{2}y + 2y] dxdy = ∫ [x^{3}y + 2xy] dy

**For “y”**

∫ [x^{3}y + 2xy] dy = ∫ [x^{3}y] dy + ∫ [2xy] dy

∫ [x^{3}y + 2xy] dy = x^{3}∫[y] dy + 2x∫ [y] dy

∫ [x^{3}y + 2xy] dy = x^{3}[y^{1+1}/1+1] + 2x [y^{1+1}/1+1] + C

∫ [x^{3}y + 2xy] dy = x^{3}[y^{2}/2] + 2x [y^{2}/2] + C

∫ [x^{3}y + 2xy] dy = x^{3}y^{2}/2 + 2xy^{2}/2 + C

∫ [x^{3}y + 2xy] dy = x^{3}y^{2}/2 + xy^{2 }+ C

**Step 4:** Final result.

∫∫ [3x^{2}y + 2y] dxdy = x^{3}y^{2}/2 + xy^{2} + C

**∫∫ [3x ^{2}y + 2y] dxdy = y^{2} [x^{3}/2 + x] + C**

Try the double integration calculator above to check the result of the above problem.

## References

- Khan Academy. (n.d.). Double integral. Khan Academy.
- Calculating double integral. Calculus III - double integrals over General Regions. (n.d.).