Step 1: Identify the integral you need to evaluate: ∫_1^2 (3x^2 - 4x + 1) dx.
Step 2: Find the antiderivative of the function 3x^2 - 4x + 1. The antiderivative is x^3 - 2x^2 + x.
Step 3: Write down the antiderivative with limits: [x^3 - 2x^2 + x] from 1 to 2.
Step 4: Calculate the value of the antiderivative at the upper limit (x = 2): 2^3 - 2(2^2) + 2 = 8 - 8 + 2 = 2.
Step 5: Calculate the value of the antiderivative at the lower limit (x = 1): 1^3 - 2(1^2) + 1 = 1 - 2 + 1 = 0.
Step 6: Subtract the lower limit result from the upper limit result: 2 - 0 = 2.
Step 7: The final answer is 2.
Definite Integral Evaluation – The process of calculating the area under a curve defined by a polynomial function over a specified interval.
Fundamental Theorem of Calculus – The theorem that connects differentiation and integration, allowing the evaluation of definite integrals using antiderivatives.
