Find the value of ∫_0^1 (1 - x^2) dx.

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Question: Find the value of ∫_0^1 (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.

Find the value of ∫_0^1 (1 - x^2) dx.

Find the value of ∫_0^1 (1 - x^2) dx.

Step 1: Identify the integral you need to solve, which is ∫_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 the first part, ∫_0^1 1 dx. The integral of 1 is x, so evaluate it from 0 to 1: [x] from 0 to 1 = 1 - 0 = 1.
Step 4: Calculate the second part, ∫_0^1 -x^2 dx. The integral of -x^2 is -x^3/3, so evaluate it from 0 to 1: [-x^3/3] from 0 to 1 = (-1/3) - (0) = -1/3.
Step 5: Combine the results from Step 3 and Step 4: 1 + (-1/3) = 1 - 1/3 = 2/3.
Step 6: The final answer for the integral ∫_0^1 (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.
Fundamental Theorem of Calculus – The solution requires applying the Fundamental Theorem of Calculus, which connects differentiation and integration.
Polynomial Integration – The integral involves integrating a polynomial function, specifically a quadratic function.