Enter the function f(x, y) to calculate the double integral (antiderivative) with this calculator.
This double integral calculator helps you evaluate definite or indefinite double integrals of two-variable functions f(x,y). Our double integral solver provides step-by-step calculations and even allows you to change the order of integration (dxdy or dydx), leading to an easier solution.
Double integral is an integration method used to find the volume under a surface, compute areas, mass, and flux over a 2D region. It integrates a function of two variables f(x, y) over a given area (denoted by R).
➥ Notation:
∫∫ R f (x, y) dA
➥ Expanded Form:
∫∫R f (x, y) dxdy
OR
∫∫R f (x, y) dydx
Where:
∫∫ = double integral
R = Region of integration in the xy-plane
f(x, y) = Function being integrated
dA = Area element which can be written as dxdy or dydx
To calculate the double integral of the 2-dimensional functions, follow these steps:
Evaluate double integral x2 + 3xy2 + xy with limit values (0, 1) for x and y variables.
Step 1: Compute The Inner Integral for variable x
\( \ ∫_{0}^{1} (x^2 + 3xy^2 + xy) \, dx \)
\(\ = \left[ \frac{x^3}{3} + \frac{3}{2}x^2y^2 + \frac{x^2}{2}y \right]_{0}^{1} \)
\(\ = \left( \frac{1^3}{3} + \frac{3}{2}(1)y^2 + \frac{1^2}{2}y \right) - \left( \frac{0^3}{3} + \frac{3}{2}(0)y^2 + \frac{0^2}{2}y \right) \)
\(\ = \left( \frac{1}{3} + \frac{3}{2}y^2 + \frac{1}{2}y \right) - 0 \) \( = \frac{1}{3} + \frac{3}{2}y^2 + \frac{1}{2}y \)
Step 2: Now integrate the result obtained in step 1 for variable y
\(\ ∫_{0}^{1} \left( \frac{1}{3} + \frac{3}{2}y^2 + \frac{1}{2}y \right) \, dy \)
\(\ = \left[ \frac{1}{3}y + \frac{1}{2}y^3 + \frac{1}{4}y^2 \right]_{0}^{1} \)
\(\ = \left( \frac{1}{3}(1) + \frac{1}{2}(1)^3 + \frac{1}{4}(1)^2 \right) - \left( \frac{1}{3}(0) + \frac{1}{2}(0)^3 + \frac{1}{4}(0)^2 \right) \)
\(\ = \left( \frac{1}{3} + \frac{1}{2} + \frac{1}{4} \right) - 0 \)
\(\ = \frac{13}{12} \)
For triple integral calculations, check our Triple Integral Calculator.
- Click the camera icon to upload an equation image or the keyboard icon for a virtual input.
- Press the "Load Example" button for a sample calculation (Optional)
Our double integration calculator uses the double integration method to handle various types of 2-dimensional functions step-wise including:
No, the order can be changed without affecting the result. According to Fubini’s Theorem, if f(x, y), if f(x, y) is continuous over a region R, then:
∫∫R f (x, y) dxdy = ∫∫R f (x, y) dydx
The double integrals calculator can not find the bounds for you. You have to define them in the case of the double definite integral; otherwise, use the indefinite integration for computing the given two-variable function without bounds.
Yes, this double integral solver allows switching the order of integration between dxdy and dydx.
This calculator is very beneficial for students studying integration and professionals associated with engineering, physics, and science fields. It is also useful for people who have to perform double integrals on two-variable functions according to their preferences.
References:
From the source of Wikipedia: Multiple integrals, Methods of integration.
From the source of libretexts.org: Double Integrals over General Regions.
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