2nd Integral Calculator

This 2nd integral calculator computes double integrals of functions with respect to two variables. It's useful for physics, engineering, and advanced mathematics applications where you need to find the volume under a surface or integrate over a region.

What is a 2nd Integral?

A second integral, also known as a double integral, is an extension of the single integral concept to two dimensions. While a single integral calculates the area under a curve, a double integral calculates the volume under a surface or the integral of a function over a region in the plane.

Double integrals are used in physics to calculate mass distributions, in engineering for stress analysis, and in probability for calculating expected values over continuous distributions.

Double integrals can be calculated using either the iterated integral method (where you integrate with respect to one variable first, then the other) or by changing to polar coordinates for certain symmetric functions.

How to Calculate 2nd Integral

The general form of a double integral is:

∫∫ f(x,y) dA = ∫[b to a] ∫[g(x) to h(x)] f(x,y) dy dx

Where:

f(x,y) is the function to be integrated
dA represents the area element
a and b are the limits of integration for x
h(x) and g(x) are the upper and lower limits of integration for y as a function of x

Step-by-Step Calculation

Identify the function f(x,y) to be integrated
Determine the region of integration in the xy-plane
Set up the iterated integral with proper limits
Integrate with respect to the inner variable first
Integrate the result with respect to the outer variable
Evaluate the definite integral

Example

Calculate ∫∫ (x² + y²) dA over the region bounded by x=0 to x=2 and y=0 to y=√(4-x²).

This represents the volume under the paraboloid z = x² + y² over a semicircle.

Example Calculation

Let's calculate the double integral of f(x,y) = x + y over the rectangle defined by 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1.

∫[0 to 2] ∫[0 to 1] (x + y) dy dx

First, integrate with respect to y:

∫[0 to 1] (x + y) dy = [xy + (y²)/2] from 0 to 1 = x(1) + (1)/2 - [x(0) + (0)/2] = x + 0.5

Then integrate with respect to x:

∫[0 to 2] (x + 0.5) dx = [(x²)/2 + 0.5x] from 0 to 2 = (4/2 + 1) - (0 + 0) = 2 + 1 = 3

The result is 3, which represents the volume under the plane z = x + y over the unit square.

FAQ

What is the difference between a single integral and a double integral?

A single integral calculates the area under a curve in two dimensions, while a double integral calculates the volume under a surface or the integral over a region in three dimensions.

When would I use a double integral in real life?

Double integrals are used in physics for mass calculations, in engineering for stress analysis, in probability for expected values, and in computer graphics for rendering.

How do I know which variable to integrate first?

The order of integration depends on the region of integration. For simple rectangular regions, either order works. For more complex regions, you may need to set up the integral in a specific order to match the region's boundaries.