Calculate ∫ from 0 to 1 of (4x^3 - 3x^2 + 2x - 1) dx.

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Question: Calculate ∫ from 0 to 1 of (4x^3 - 3x^2 + 2x - 1) dx.

Options:

Correct Answer: 1

Solution:

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

Calculate ∫ from 0 to 1 of (4x^3 - 3x^2 + 2x - 1) dx.

Calculate ∫ from 0 to 1 of (4x^3 - 3x^2 + 2x - 1) dx.

Correct Answer: 0

Step 1: Identify the function to integrate, which is (4x^3 - 3x^2 + 2x - 1).
Step 2: Find the antiderivative of the function. This means we need to integrate each term separately.
Step 3: Integrate 4x^3 to get x^4.
Step 4: Integrate -3x^2 to get -x^3.
Step 5: Integrate 2x to get x^2.
Step 6: Integrate -1 to get -x.
Step 7: Combine all the antiderivatives to get the complete antiderivative: x^4 - x^3 + x^2 - x.
Step 8: Evaluate the antiderivative from 0 to 1. This means we will calculate the value at 1 and then subtract the value at 0.
Step 9: Calculate the value at 1: (1^4 - 1^3 + 1^2 - 1) = (1 - 1 + 1 - 1) = 0.
Step 10: Calculate the value at 0: (0^4 - 0^3 + 0^2 - 0) = 0.
Step 11: Subtract the value at 0 from the value at 1: 0 - 0 = 0.

Definite Integral – The process of calculating the area under a curve defined by a polynomial function over a specified interval.
Polynomial Integration – Applying the power rule for integration to find the antiderivative of polynomial terms.
Fundamental Theorem of Calculus – Connecting differentiation and integration, allowing evaluation of definite integrals using antiderivatives.