What is the value of the integral ∫(0 to 1) x^2 dx?

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Question: What is the value of the integral ∫(0 to 1) x^2 dx?

Options:

Correct Answer: 1/3

Solution:

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

What is the value of the integral ∫(0 to 1) x^2 dx?

What is the value of the integral ∫(0 to 1) x^2 dx?

Step 1: Identify the integral you need to solve, which is ∫(0 to 1) x^2 dx.
Step 2: Find the antiderivative of x^2. The antiderivative is (x^3)/3.
Step 3: Write down the antiderivative with limits: [(x^3)/3] from 0 to 1.
Step 4: Substitute the upper limit (1) into the antiderivative: (1^3)/3 = 1/3.
Step 5: Substitute the lower limit (0) into the antiderivative: (0^3)/3 = 0.
Step 6: Subtract the value from the lower limit from the value from the upper limit: (1/3) - (0) = 1/3.
Step 7: Conclude that the value of the integral ∫(0 to 1) x^2 dx is 1/3.

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