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

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

Options:

Correct Answer: 1/2

Solution:

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

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

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

Correct Answer: 2/3

Step 1: Identify the integral you need to calculate: ∫ from 0 to 1 of (1 - x^2) dx.
Step 2: Find the antiderivative of the function (1 - x^2). The antiderivative is x - (x^3)/3.
Step 3: Write down the antiderivative: F(x) = x - (x^3)/3.
Step 4: Evaluate the antiderivative at the upper limit (1): F(1) = 1 - (1^3)/3 = 1 - 1/3 = 2/3.
Step 5: Evaluate the antiderivative at the lower limit (0): F(0) = 0 - (0^3)/3 = 0.
Step 6: Subtract the value at the lower limit from the value at the upper limit: (2/3) - (0) = 2/3.
Step 7: Conclude that the value of the integral ∫ from 0 to 1 of (1 - x^2) dx is 2/3.

Definite Integral – The question tests the ability to evaluate a definite integral, which involves finding the area under the curve of the function from the lower limit to the upper limit.
Polynomial Integration – The integral involves a polynomial function, specifically a quadratic function, which requires knowledge of basic integration rules.