Calculate ∫_0^1 (x^3 - 2x^2 + x) dx.

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Question: Calculate ∫_0^1 (x^3 - 2x^2 + x) dx.

Options:

Correct Answer: 1/12

Solution:

The integral evaluates to [x^4/4 - 2x^3/3 + x^2/2] from 0 to 1 = (1/4 - 2/3 + 1/2) = 1/12.

Calculate ∫_0^1 (x^3 - 2x^2 + x) dx.

Calculate ∫_0^1 (x^3 - 2x^2 + x) dx.

Step 1: Identify the function to integrate, which is f(x) = x^3 - 2x^2 + x.
Step 2: Find the antiderivative of f(x). This means we need to calculate the integral of each term separately.
Step 3: For the term x^3, the antiderivative is (x^4)/4.
Step 4: For the term -2x^2, the antiderivative is -2(x^3)/3.
Step 5: For the term x, the antiderivative is (x^2)/2.
Step 6: Combine the antiderivatives to get the complete antiderivative: (x^4)/4 - (2x^3)/3 + (x^2)/2.
Step 7: Now evaluate this antiderivative from 0 to 1. This means we will calculate it at x = 1 and then subtract the value at x = 0.
Step 8: Calculate the value at x = 1: (1^4)/4 - (2*1^3)/3 + (1^2)/2 = 1/4 - 2/3 + 1/2.
Step 9: Calculate the value at x = 0: (0^4)/4 - (2*0^3)/3 + (0^2)/2 = 0.
Step 10: Subtract the value at x = 0 from the value at x = 1: (1/4 - 2/3 + 1/2) - 0 = 1/4 - 2/3 + 1/2.
Step 11: Simplify the expression: Convert all terms to have a common denominator (12): 3/12 - 8/12 + 6/12 = 1/12.

Definite Integrals – The question tests the ability to compute a definite integral of a polynomial function over a specified interval.
Polynomial Integration – It assesses the understanding of integrating polynomial functions term by term.
Evaluation of Limits – The question requires evaluating the antiderivative at the upper and lower limits of integration.