Calculate The Double Integral.r5xx2 Y2da R 1 2 0 1

This guide explains how to calculate the double integral of the function 5x² + 2y² over the rectangular region from x=0 to x=1 and y=0 to y=2. We'll cover the mathematical process, provide a calculator, and include practical examples.

What is a double integral?

A double integral extends the concept of single integration to two dimensions. It calculates the volume under a surface defined by a function of two variables over a region in the xy-plane. For a function f(x,y), the double integral is written as ∫∫ f(x,y) dA.

Double integrals are used in physics, engineering, and mathematics to calculate quantities like mass, probability, and work over two-dimensional regions. The most common method for evaluating double integrals is iterated integration, where we integrate with respect to one variable first and then the other.

How to calculate the double integral

To calculate ∫∫ (5x² + 2y²) dA from 0 to 1 and 0 to 2, follow these steps:

Identify the limits of integration: x from 0 to 1, y from 0 to 2
Split the integrand into two parts: 5x² and 2y²
Integrate each part separately with respect to y first, then x
Combine the results to get the final volume

The general formula for this calculation is:

∫₀¹ ∫₀² (5x² + 2y²) dy dx

Step-by-step calculation

First integrate with respect to y:
∫₀² (5x² + 2y²) dy = 5x²y + (2/3)y³ evaluated from 0 to 2

= [5x²(2) + (2/3)(8)] - [0 + 0] = 10x² + (16/3)
Now integrate the result with respect to x:
∫₀¹ (10x² + 16/3) dx = (10/3)x³ + (16/3)x evaluated from 0 to 1

= [(10/3)(1) + (16/3)(1)] - [0 + 0] = (10/3) + (16/3) = 26/3

Example calculation

Let's calculate the double integral for the function 5x² + 2y² over the region x=0 to x=1 and y=0 to 2:

The result of this calculation is 26/3, which is approximately 8.6667.

This means the volume under the surface defined by 5x² + 2y² over the given region is 26/3 cubic units.

FAQ

What is the difference between single and double integrals?: A single integral calculates area under a curve, while a double integral calculates volume under a surface over a two-dimensional region.
When would I use a double integral in real life?: Double integrals are used in physics to calculate mass distributions, in engineering for calculating moments of inertia, and in probability for calculating joint probabilities.
Can I calculate double integrals for non-rectangular regions?: Yes, but it requires more advanced techniques like polar or cylindrical coordinates, or using Green's theorem for certain regions.
What if my function is more complex than 5x² + 2y²?: The same process applies - split the integrand into simpler terms, integrate with respect to y first, then x, and combine the results.