Evaluate ∫_0^1 (1 - x^2) dx.

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Question: Evaluate ∫_0^1 (1 - x^2) dx.

Options:

Correct Answer: 1/2

Solution:

∫_0^1 (1 - x^2) dx = [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.

Evaluate ∫_0^1 (1 - x^2) dx.

Evaluate ∫_0^1 (1 - x^2) dx.

Step 1: Identify the integral you need to evaluate: ∫_0^1 (1 - x^2) dx.
Step 2: Break down the integral into two parts: ∫_0^1 1 dx and ∫_0^1 -x^2 dx.
Step 3: Calculate ∫_0^1 1 dx. This is simply the area under the curve from 0 to 1, which equals 1.
Step 4: Calculate ∫_0^1 -x^2 dx. The antiderivative of -x^2 is -x^3/3.
Step 5: Evaluate the antiderivative from 0 to 1: [-x^3/3] from 0 to 1 = [-(1^3)/3 - (0^3)/3] = -1/3.
Step 6: Combine the results from Step 3 and Step 5: 1 + (-1/3) = 1 - 1/3 = 2/3.
Step 7: Write the final answer: ∫_0^1 (1 - x^2) dx = 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.
Fundamental Theorem of Calculus – The solution requires applying the Fundamental Theorem of Calculus, which connects differentiation and integration, allowing the evaluation of the integral using antiderivatives.
Polynomial Integration – The integral involves a polynomial function, testing the knowledge of integrating polynomial expressions.