Calculate The Double Integral R 5x X2 Y2 Da

This guide explains how to calculate the double integral ∫∫R 5x x² y² dA, including the formula, assumptions, and practical applications. The interactive calculator on this page makes it easy to compute the integral for different regions and functions.

What is a double integral?

A double integral extends the concept of single integration to two dimensions. It calculates the volume under a surface z = f(x,y) over a region R in the xy-plane. The double integral is written as ∫∫R f(x,y) dA, where dA represents an infinitesimal area element.

Double integrals have applications in physics, engineering, and economics, including calculating mass distributions, fluid flow, and probability densities. The integral can be computed using either Cartesian coordinates or polar coordinates, depending on the shape of the region R.

Calculating the double integral

Formula

Double Integral Formula

∫∫R f(x,y) dA = ∫[a,b] ∫[g1(x),g2(x)] f(x,y) dy dx

For the given integral ∫∫R 5x x² y² dA, we'll use Cartesian coordinates with limits of integration based on the region R.

Steps to calculate

Identify the region R and determine the limits of integration for x and y.
Integrate the function with respect to y first, treating x as a constant.
Integrate the resulting expression with respect to x.
Evaluate the definite integral using the specified limits.

Assumptions

This calculation assumes the region R is defined by x from a to b and y from g1(x) to g2(x). The function f(x,y) = 5x x² y² is continuous over the region R.

Example calculation

Let's compute the double integral ∫∫R 5x x² y² dA over the region R defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ x.

Step 1: Set up the integral

∫[0,1] ∫[0,x] 5x x² y² dy dx

Step 2: Integrate with respect to y

∫[0,x] 5x x² y² dy = 5x³ [y³/3] from 0 to x = 5x³ (x³/3) = (5/3)x⁶

Step 3: Integrate with respect to x

∫[0,1] (5/3)x⁶ dx = (5/3) [x⁷/7] from 0 to 1 = (5/3)(1/7) = 5/21

Result

The value of the double integral is 5/21.

FAQ

What is the difference between single and double integrals?

A single integral calculates the area under a curve in one dimension, while a double integral calculates the volume under a surface in two dimensions. Double integrals require limits of integration for both x and y.

When would I use a double integral in real life?

Double integrals are used in physics to calculate mass distributions, in engineering to find centroids, and in probability to compute expected values over two-dimensional regions.

How do I choose between Cartesian and polar coordinates for double integrals?

Use Cartesian coordinates when the region R is defined by vertical or horizontal boundaries, and polar coordinates when the region is circular or has radial symmetry.