Evaluate the integral ∫(0 to 1) (1 - x^2) dx. (2022)

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Question: Evaluate the integral ∫(0 to 1) (1 - x^2) dx. (2022)

Options:

Correct Answer: 2/3

Exam Year: 2022

Solution:

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

Evaluate the integral ∫(0 to 1) (1 - x^2) dx. (2022)

Evaluate the integral ∫(0 to 1) (1 - x^2) dx. (2022)

Step 1: Identify the integral you need to evaluate: ∫(0 to 1) (1 - x^2) dx.
Step 2: Break down the integrand (1 - x^2) into two separate parts: 1 and -x^2.
Step 3: Find the antiderivative of each part: The antiderivative of 1 is x, and the antiderivative of -x^2 is -x^3/3.
Step 4: Combine the antiderivatives: The complete antiderivative is x - x^3/3.
Step 5: Evaluate the antiderivative from the lower limit (0) to the upper limit (1): Substitute 1 into the antiderivative: 1 - (1^3)/3 = 1 - 1/3.
Step 6: Substitute 0 into the antiderivative: 0 - (0^3)/3 = 0.
Step 7: Calculate the definite integral: (1 - 1/3) - 0 = 1 - 1/3 = 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 (1 - x^2), requiring knowledge of basic integration techniques for polynomials.
Fundamental Theorem of Calculus – The solution applies the Fundamental Theorem of Calculus, which connects differentiation and integration, allowing for the evaluation of definite integrals using antiderivatives.