Calculate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.

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Question: Calculate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.

Options:

Correct Answer: 6

Solution:

The integral evaluates to [x^4/4 - x^3 + 4x] from 0 to 2 = (4 - 8 + 8) - 0 = 4.

Calculate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.

Calculate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.

Step 1: Identify the function to integrate, which is f(x) = x^3 - 3x^2 + 4.
Step 2: Find the antiderivative (indefinite integral) of f(x). This means we need to calculate ∫(x^3 - 3x^2 + 4) dx.
Step 3: Integrate each term separately: ∫x^3 dx = x^4/4, ∫(-3x^2) dx = -x^3, and ∫4 dx = 4x.
Step 4: Combine the results of the integration: The antiderivative is F(x) = x^4/4 - x^3 + 4x + C, where C is the constant of integration.
Step 5: Evaluate the definite integral from 0 to 2. This means we need to calculate F(2) - F(0).
Step 6: Calculate F(2): F(2) = (2^4)/4 - (2^3) + 4(2) = 16/4 - 8 + 8 = 4.
Step 7: Calculate F(0): F(0) = (0^4)/4 - (0^3) + 4(0) = 0.
Step 8: Subtract the two results: F(2) - F(0) = 4 - 0 = 4.

Definite Integral – The process of calculating the area under a curve defined by a function over a specific interval.
Fundamental Theorem of Calculus – Relates differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.
Polynomial Integration – Involves integrating polynomial functions term by term.