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

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

Options:

Correct Answer: 1

Solution:

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

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

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

Correct Answer: 2/3

Step 1: Identify the function to integrate, which is (4x^3 - 4x^2 + 1).
Step 2: Find the antiderivative of the function. The antiderivative is x^4 - (4/3)x^3 + x.
Step 3: Evaluate the antiderivative at the upper limit (1) and the lower limit (0).
Step 4: Calculate the value at the upper limit: (1^4 - (4/3)(1^3) + 1) = 1 - (4/3) + 1.
Step 5: Calculate the value at the lower limit: (0^4 - (4/3)(0^3) + 0) = 0.
Step 6: Subtract the lower limit value from the upper limit value: (1 - (4/3) + 1) - 0 = 2 - (4/3).
Step 7: Simplify the result: 2 - (4/3) = (6/3) - (4/3) = (2/3).

Definite Integral – Calculating the area under the curve of a polynomial function over a specified interval.
Polynomial Integration – Applying the power rule for integration to polynomial terms.
Fundamental Theorem of Calculus – Using the theorem to evaluate the definite integral by finding the antiderivative.