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

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

Options:

Correct Answer: 2/3

Solution:

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

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

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

Step 1: Identify the integral you need to solve: ∫(0 to 1) (1 - x^2)dx.
Step 2: Break down the integral into two parts: ∫(0 to 1) 1 dx and ∫(0 to 1) x^2 dx.
Step 3: Calculate the first part: ∫(0 to 1) 1 dx = [x] from 0 to 1 = 1 - 0 = 1.
Step 4: Calculate the second part: ∫(0 to 1) x^2 dx = [(1/3)x^3] from 0 to 1 = (1/3)(1^3) - (1/3)(0^3) = 1/3 - 0 = 1/3.
Step 5: Combine the results from Step 3 and Step 4: 1 - 1/3.
Step 6: Simplify the result: 1 - 1/3 = 3/3 - 1/3 = 2/3.

Definite Integral – The process of calculating the area under a curve defined by a function over a specified interval.
Polynomial Integration – Applying the power rule for integration to polynomials, which involves increasing the exponent and dividing by the new exponent.