Double Integral Calculator

Double Integral?

In Calculus, a double crucial is used to compute the integrals of variable capabilities ( denoted by way of f(x, y)) over a -dimensional place (denoted by means of R). It now not simplest enables to locate the volume underneath surfaces however also the mass distribution, and compute flux (fee of drift) and area over the vicinity\(\ R^{2}\ \).

A double quintessential is mathematically represented via the symbol \(\ ”∫∫_R”\), which shows a double quintessential over area "R" observed by the characteristic f(x, y) and the area detail dA.

A double necessary also can be shown as an iterated vital:

\(\begin{array}{l}\ ∫∫_{R}f(x,y)\ dA =\ ∫∫_{R}f(x,y)\ dx\ dy\end{array}\)

Example:

Evaluate double integral \(\ x^{2}\ + \ 3xy^{2}\ + \ xy\) with limit values (0, 2) for x and y variables.

Solution:

Step 1: Compute The Inner Integral for variable x

 \( \ ∫_{0}^{2} (x^2 + 3xy^2 + xy) \, dx \)

 

\(\ = \left[ \frac{x^3}{3} + \frac{3}{2}x^2y^2 + \frac{x^2}{2}y \right]_{0}^{2} \)

 

\(\ = \left( \frac{2^3}{3} + \frac{3}{2}(2)^2y^2 + \frac{2^2}{2}y \right) - \left( \frac{0^3}{3} + \frac{3}{2}(0)^2y^2 + \frac{0^2}{2}y \right) \)

 

\(\ = \left( \frac{8}{3} + \frac{3}{2}(4)y^2 + \frac{4}{2}y \right) - 0 \) \( = \frac{8}{3} + 6y^2 + 2y \)

 

Step 2: Now integrate the end result received in step 1 for variable y

 

\(\ ∫_{0}^{2} \left( \frac{8}{3} + 6y^2 + 2y \right) \, dy \)

 

\(\ = \left[ \frac{8}{3}y + 2y^3 + y^2 \right]_{0}^{2} \)

 

\(\ = \left( \frac{8}{3}(2) + 2(2)^3 + (2)^2 \right) - \left( \frac{8}{3}(0) + 2(0)^3 + (0)^2 \right) \)

 

\(\ = \left( \frac{16}{3} + 16 + 4 \right) - 0 \)

 

\(\ = \frac{16}{3} + 20 \)

 

\(\ = \frac{16}{3} + \frac{60}{3} = \frac{76}{3} \)

A way to Use The Double vital Calculator?

• Enter a function which you need to integrate and choose its integration order
• Choose the choice whether you need to make particular or indefinite integration with this calculator and values as a consequence to proceed with the integration method
• click on at the “Calculate Double integral” button to assess the double integral step-sensible

FAQ’s:

What if I'm unsure about the boundaries of integration?

The double integrals calculator can not locate the bounds for you. you have to outline them within the case of the double exact vital, in any other case use the indefinite integration for computing the given -variable feature without bounds.

Am i able to reverse the order of integration with this calculator?

sure, the calculator for double integration can opposite the order of integration. It includes converting the order in which the integration is carried out for two variables (x, y).

Is it viable to split the double necessary?

sure, double integrals are solved with iterated integration strategies (or repeated integration). It does now not directly "split up" the double indispensable, but rather breaks it into nested integrals, one for each imperative. also, the iterated fundamental calculator is a possible manner to combine a function of a couple of variables.