Calculate The Double Integral 3xy2 X2 1 Da

This guide explains how to calculate the double integral of 3xy² over the region defined by x² to 1 in the da plane. 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. The double integral of a function f(x,y) over a region R is written as:

∫∫_R f(x,y) dA

For this calculation, we're evaluating the integral of 3xy² over the region where x ranges from x² to 1.

Key concepts

The integrand is 3xy²
The region of integration is defined by x² ≤ x ≤ 1
dA represents the differential area element

How to calculate this double integral

The calculation involves setting up the integral with proper limits and performing the integration step by step.

Step 1: Set up the integral

First, express the double integral with the given limits:

∫_x=x²¹ ∫_y=0^x 3xy² dy dx

Step 2: Integrate with respect to y

Perform the inner integration (with respect to y) first:

∫_y=0^x 3xy² dy = 3x ∫₀^x y² dy = 3x [y³/3]₀ˣ = 3x (x³/3) = x⁴

Step 3: Integrate with respect to x

Now integrate the result with respect to x:

∫_x=x²¹ x⁴ dx = [x⁵/5]ₓ²¹ = (1⁵/5) - (x²)⁵/5 = 1/5 - x¹⁰/5

Step 4: Evaluate the definite integral

Substitute the limits of integration to find the final value.

Note: The exact value depends on the specific limits of integration. The calculator below allows you to input your own limits if needed.

Example calculation

Let's work through an example with specific limits to see how this calculation works in practice.

Example problem

Calculate the double integral of 3xy² over the region where x ranges from 0 to 1 and y ranges from 0 to x.

Step-by-step solution

Set up the integral:
∫₀¹ ∫₀^x 3xy² dy dx
Integrate with respect to y:
∫₀^x 3xy² dy = x⁴
Integrate with respect to x:
∫₀¹ x⁴ dx = 1/5

The result of this specific example is 1/5. The calculator below can handle different limits if needed.

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 region in two dimensions.
How do I know when to use a double integral?: Use double integrals when you need to calculate quantities that depend on two variables, such as mass, probability, or volume in 3D space.
What if my limits of integration are different?: The calculator below allows you to input your own limits. The general approach remains the same: integrate with respect to the inner variable first, then the outer variable.
Can I visualize the region of integration?: The calculator includes a visualization of the region when possible, helping you understand the limits of integration.