Find the value of ∫ from 0 to 1 of (1 - x^2) dx.

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

Options:

Correct Answer: 2/3

Solution:

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

Find the value of ∫ from 0 to 1 of (1 - x^2) dx.

Find the value of ∫ from 0 to 1 of (1 - x^2) dx.

Step 1: Identify the integral you need to solve: ∫ from 0 to 1 of (1 - x^2) dx.
Step 2: Break down the integral into two parts: ∫ from 0 to 1 of 1 dx and ∫ from 0 to 1 of x^2 dx.
Step 3: Calculate the first part: ∫ from 0 to 1 of 1 dx = [x] from 0 to 1 = 1 - 0 = 1.
Step 4: Calculate the second part: ∫ from 0 to 1 of x^2 dx = [x^3/3] from 0 to 1 = (1^3/3) - (0^3/3) = 1/3 - 0 = 1/3.
Step 5: Combine the results from Step 3 and Step 4: 1 - 1/3 = 2/3.
Step 6: Write the final answer: The value of the integral 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, requiring knowledge of basic integration rules for polynomials.