Calculate The Iterated Integral 6x2y 4x Dy Dx

This guide explains how to calculate the double integral ∫∫(6x²y + 4x) dy dx over a specified region. We'll cover the mathematical approach, provide a step-by-step calculation method, and include an interactive calculator to compute the result for any given limits.

How to Calculate the Iterated Integral

The iterated integral ∫∫(6x²y + 4x) dy dx represents a double integral of the function 6x²y + 4x with respect to y first, then x. To evaluate this, we follow these steps:

General Form: ∫_a^b ∫_c^d (6x²y + 4x) dy dx

The calculation involves integrating the inner function with respect to y first, then integrating the result with respect to x. The limits of integration (a, b, c, d) depend on the region over which you're integrating.

Key Concepts

The order of integration (dy dx) indicates we integrate with respect to y first, then x
The integrand is a function of both x and y
The limits of integration define the region of integration

Note: For this calculation, we'll assume the limits of integration are from x=0 to x=2 and y=0 to y=x², creating a triangular region in the xy-plane.

Step-by-Step Calculation

Let's calculate ∫₀² ∫₀^x² (6x²y + 4x) dy dx step by step.

Step 1: Integrate with respect to y

First, integrate the integrand with respect to y while treating x as a constant:

∫(6x²y + 4x) dy = 6x²∫y dy + 4x∫1 dy = 6x²(y²/2) + 4x(y) + C = 3x²y² + 4xy + C

Step 2: Apply the y limits

Now evaluate the antiderivative at the upper and lower y limits (0 to x²):

[3x²y² + 4xy] from y=0 to y=x² = 3x²(x²)² + 4x(x²) - [3x²(0)² + 4x(0)] = 3x⁶ + 4x³

Step 3: Integrate with respect to x

Now integrate the result from Step 2 with respect to x:

∫(3x⁶ + 4x³) dx = 3(x⁷/7) + 4(x⁴/4) + C = (3/7)x⁷ + x⁴ + C

Step 4: Apply the x limits

Finally, evaluate the antiderivative at the upper and lower x limits (0 to 2):

[(3/7)(2)⁷ + (2)⁴] - [(3/7)(0)⁷ + (0)⁴] = (3/7)(128) + 16 = 57.1429 + 16 = 73.1429

Worked Example

Let's calculate ∫₀² ∫₀^x² (6x²y + 4x) dy dx using the steps above.

Example Calculation

Step 1: ∫(6x²y + 4x) dy = 3x²y² + 4xy

Step 2: [3x²y² + 4xy] from y=0 to y=x² = 3x⁶ + 4x³

Step 3: ∫(3x⁶ + 4x³) dx = (3/7)x⁷ + x⁴

Step 4: [(3/7)x⁷ + x⁴] from x=0 to x=2 = 73.1429

This example shows how to compute the double integral for specific limits. The calculator on the right allows you to compute this for any limits you specify.

Frequently Asked Questions

What is the difference between single and double integrals?

A single integral calculates area under a curve in one dimension, while a double integral calculates volume under a surface in two dimensions. Double integrals require integration with respect to two variables.

How do I choose the order of integration?

The order of integration depends on the region of integration. For simple regions like triangles or rectangles, you can choose either order. For more complex regions, you may need to sketch the region to determine the correct order.

What are the limits of integration for this problem?

The limits depend on the region you're integrating over. In our example, we used x from 0 to 2 and y from 0 to x², creating a triangular region. You can change these limits in the calculator to match your specific problem.