Step 1: Identify the integral to be calculated: ∫_0^1 (4x^3 - 3x^2 + 2) dx.
Step 2: Find the antiderivative of the function (4x^3 - 3x^2 + 2).
Step 3: The antiderivative is calculated as follows: For 4x^3, the antiderivative is (4/4)x^4 = x^4; for -3x^2, it is (-3/3)x^3 = -x^3; and for 2, it is 2x.
Step 4: Combine the antiderivatives: The complete antiderivative is x^4 - x^3 + 2x.
Step 5: Evaluate the antiderivative from 0 to 1: Substitute 1 into the antiderivative: (1^4 - 1^3 + 2*1) = (1 - 1 + 2) = 2.
Step 6: Substitute 0 into the antiderivative: (0^4 - 0^3 + 2*0) = (0 - 0 + 0) = 0.
Step 7: Subtract the value at 0 from the value at 1: 2 - 0 = 2.
Step 8: The final answer is 2.
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.
