Calculate The Iterated Integral.101 2x Y 4 Dx Dy

This guide explains how to calculate the iterated integral ∫∫ (2x + y^4) dx dy, including the step-by-step process, formula explanation, and practical examples. The interactive calculator on this page makes it easy to compute the integral for any given limits.

What is an iterated integral?

An iterated integral is a double integral where one integral is evaluated first, followed by another integral. The order of integration (dx dy or dy dx) affects the result. For ∫∫ (2x + y^4) dx dy, we integrate with respect to x first, then y.

Key Points

Iterated integrals are evaluated in sequence
The order of integration matters (dx dy vs dy dx)
Requires defining limits for both variables
Used in physics, engineering, and calculus problems

How to calculate ∫∫ (2x + y^4) dx dy

To compute the iterated integral ∫∫ (2x + y^4) dx dy, follow these steps:

Define the limits of integration for x and y
Integrate the integrand with respect to x first
Integrate the result with respect to y
Evaluate the definite integral using the given limits

Formula

∫_a^b ∫_c^d (2x + y^4) dx dy = ∫_a^b [∫_c^d (2x + y^4) dx] dy

First integral (with respect to x): ∫ (2x + y^4) dx = x² + x y⁴ + C

Second integral (with respect to y): ∫ (x² + x y⁴ + C) dy = x² y + (x y⁵)/5 + C'

After integrating, evaluate the definite integral using the specified limits for x and y. The calculator on this page automates this process for any given limits.

Example calculation

Let's compute ∫₀¹ ∫₀¹ (2x + y^4) dx dy:

First integral: ∫ (2x + y^4) dx = x² + x y⁴
Evaluate from x=0 to x=1: (1² + 1·y⁴) - (0² + 0·y⁴) = 1 + y⁴
Second integral: ∫ (1 + y⁴) dy = y + y⁵/5
Evaluate from y=0 to y=1: (1 + 1⁵/5) - (0 + 0⁵/5) = 1 + 1/5 = 1.2

Result

The value of ∫₀¹ ∫₀¹ (2x + y^4) dx dy is:

1.2

FAQ

What's the difference between dx dy and dy dx?

The order of integration (dx dy vs dy dx) affects the result. For ∫∫ (2x + y^4) dx dy, we integrate with respect to x first, then y. Changing the order would require adjusting the limits of integration accordingly.

Can I use the calculator for any limits?

Yes, the calculator on this page accepts any numerical limits for x and y. Simply enter your desired limits and click "Calculate" to get the result.

What if my integrand is more complex?

This calculator is specifically designed for the integrand (2x + y^4). For more complex integrands, you may need specialized mathematical software or advanced calculus techniques.