Question

The value of the integral \( \int_0^1 x^2 \, dx \) is:

• A. \( \frac{1}{3} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{2}{3} \)
• D. \( 1 \)

Hint:

When integrating polynomials, use the power rule and evaluate the result by substituting the upper and lower limits of the integral.

Answer 1

The Correct Option is A

We are asked to find the value of the integral \( \int_0^1 x^2 \, dx \). Step 1: Use the power rule of integration We know the power rule of integration: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] For our integral, we have \( n = 2 \). \[ \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 \] Step 2: Evaluate the integral Now, substitute the limits of integration: \[ = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} - 0 = \frac{1}{3} \] Answer: The value of the integral is \( \frac{1}{3} \), so the correct answer is option (1).