Calcula La Integral: Integral 3 X Al Cuadrado D X

How to Calculate the Integral of 3x² dx

To calculate the integral of 3x² dx, you'll need to follow these steps:

1. Identify the function to be integrated: f(x) = 3x².
2. Apply the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration.
3. Multiply the coefficient (3) by the result of the integration.
4. Add the constant of integration (C) to the result.

The result of this integration will give you the antiderivative of 3x², which represents the family of curves whose derivative is 3x².

The Formula for Integration

For the specific case of 3x² dx, we apply the power rule with n = 2:

∫3x² dx = 3 * (x³/3) + C = x³ + C

Worked Example

Let's calculate the integral of 3x² dx from x = 0 to x = 2:

1. First, find the antiderivative: ∫3x² dx = x³ + C
2. Evaluate the antiderivative at the upper limit (x = 2): (2)³ = 8
3. Evaluate the antiderivative at the lower limit (x = 0): (0)³ = 0
4. Subtract the lower limit evaluation from the upper limit evaluation: 8 - 0 = 8

The definite integral of 3x² dx from 0 to 2 is 8. This means the area under the curve of f(x) = 3x² between x = 0 and x = 2 is 8 square units.

Practical Applications

Understanding how to calculate the integral of 3x² dx has several practical applications:

• In physics, integrals are used to calculate areas under velocity-time graphs to determine displacement.
• In engineering, integrals help determine the center of mass of objects with varying densities.
• In economics, integrals are used to calculate total revenue, total cost, and total profit functions.

By mastering this fundamental calculus operation, you'll be better equipped to solve more complex problems in various fields.