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Double Integral Calculator
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Double Integrals are most widely used to calculate the area of a two dimensional figures. The first integral includes the boundary values of x and the second integral includes the boundary values of y. The boundary values of y is taken as the first preference if we take dydx and similarly the boundary value of x is taken as the first preference if we take dxdy.

Below is given a default function with its limits, click "Calculate". You can see that integration of inner integral i.e. x takes place with respect to its boundary values. After getting the integrated value, integration of outer integral i.e. y takes place with respect to its boundary values.

## Steps to Find Double Integral

Step 1 :

Observe the given function and limits for given varibales.

Step 2 :

Check the boundary values of inner integral and integrate accordingly.

Step 3 :

After getting the integrated value of the inner inner integral, start integrating the outer integral with their respective boundary values.

## Double Integral Problems

1. ### Evaluate $\int_{x=0}^{1}\int_{y=0}^{2} xy\ dy dx$

Step 1 :

Let I = $\int_{x=0}^{1}\int_{y=0}^{2} xy\ dy dx$

Step 2 :

I = $\int_{x=0}^{1}\left[\int_{y=0}^{2} xy\ dy \right] dx$
I = $\int_{x=0}^{1}\left[x \frac{y^{2}}{2}\right]_{0}^{2} dx$
I = $\int_{x=0}^{1} x [2 - 0] dx$

Step 3 :

I = $\int_{x=0}^{1} 2x dx$
I = 2 $\int_{x=0}^{1} x dx$
I = 2 $\left[\frac{x^{2}}{2}\right]_{0}^{1}$
I = $1^{2}$-0

I = 1

2. ### Evaluate $\int_{y = 0}^{2} \int_{x = 0}^{1}[x - y]\ dx dy$

Step 1 :

Let I = $\int_{y = 0}^{2} \int_{x = 0}^{1}[x - y]\ dx dy$

Step 2 :

I = $\int_{y = 0}^{2} \left[\int_{x = 0}^{1}[x - y]\ dx \right]dy$
I = $\int_{y = 0}^{2} \left[\frac{x^{2}}{2} - xy \right]_{x = 0}^{1}dy$

Step 3 :

I = $\int_{y = 0}^{2} \left[\frac{1}{2} - y \right]dy$
I = $\left[\frac{1}{2}y - \frac{y^{2}}{2} \right]_{y = 0}^{2}$
I = $\left[\frac{2}{2} - \frac{2^{2}}{2} \right]$
I = $\left[1 - \frac{4}{2} \right]$
I = $\frac{2 - 4}{2}$
I = $\frac{-2}{2}$
I = -1