Double Integral Calculator

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 R². 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) = 3x2y + 2y

Solution

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

∫∫ f(x, y) dxdy = ∫∫ [3x²y + 2y] dxdy

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

∫∫ [3x²y + 2y] dxdy = ∫ [ ∫[3x²y + 2y] dx] dy …. (1)

For “x”

∫[3x²y + 2y] dx = ∫[3x²y] dx + ∫[2y] dx

∫[3x²y + 2y] dx = 3y∫[x²] dx + 2y∫[1] dx

∫[3x²y + 2y] dx = 3y [x²⁺¹/2+1] + 2y[x]

∫[3x²y + 2y] dx = 3y [x³/3] + 2y[x]

∫[3x²y + 2y] dx = 3x³y/3 + 2xy

∫[3x²y + 2y] dx = x³y + 2xy

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

∫∫ [3x²y + 2y] dxdy = ∫ [x³y + 2xy] dy

For “y”

∫ [x³y + 2xy] dy = ∫ [x³y] dy + ∫ [2xy] dy

∫ [x³y + 2xy] dy = x³∫[y] dy + 2x∫ [y] dy

∫ [x³y + 2xy] dy = x³[y¹⁺¹/1+1] + 2x [y¹⁺¹/1+1] + C

∫ [x³y + 2xy] dy = x³[y²/2] + 2x [y²/2] + C

∫ [x³y + 2xy] dy = x³y²/2 + 2xy²/2 + C

∫ [x³y + 2xy] dy = x³y²/2 + xy² + C

Step 4: Final result.

∫∫ [3x²y + 2y] dxdy = x³y²/2 + xy² + C

∫∫ [3x²y + 2y] dxdy = y² [x³/2 + 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.).