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

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

Step 1: Identify the integral you need to evaluate: ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.
Step 2: Find the antiderivative of the function (x^3 - 3x^2 + 4).
Step 3: The antiderivative is calculated as follows: For x^3, the antiderivative is x^4/4; for -3x^2, it is -x^3; and for 4, it is 4x.
Step 4: Combine these results to get the complete antiderivative: (x^4/4 - x^3 + 4x).
Step 5: Now, evaluate this antiderivative from 0 to 2. First, substitute 2 into the antiderivative: (2^4/4 - 2^3 + 4*2).
Step 6: Calculate 2^4/4, which is 16/4 = 4; then calculate -2^3, which is -8; and finally calculate 4*2, which is 8.
Step 7: Combine these results: 4 - 8 + 8 = 4.
Step 8: Now, substitute 0 into the antiderivative: (0^4/4 - 0^3 + 4*0), which equals 0.
Step 9: Finally, subtract the value at 0 from the value at 2: 4 - 0 = 4.

Definite Integral Evaluation – 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.