Question

Find the value of \( \int_a^b x^2 \, dx \) with the help of definite integral as the limit of a sum.

Hint:

To evaluate definite integrals using the limit of a sum, express the integral as a Riemann sum and take the limit as the number of subintervals approaches infinity.

Answer 1

We are asked to evaluate the definite integral \( \int_a^b x^2 \, dx \) using the limit of a sum. The definition of a definite integral is: \[ \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x, \] where \( \Delta x = \frac{b - a}{n} \) is the width of each subinterval and \( x_i \) are sample points in the subintervals. For the function \( f(x) = x^2 \), the Riemann sum is: \[ \sum_{i=1}^{n} \left( \frac{a + i \Delta x}{n} \right)^2 \Delta x. \] Thus, the integral becomes: \[ \lim_{n \to \infty} \sum_{i=1}^{n} \left( \frac{a + i \Delta x}{n} \right)^2 \Delta x = \lim_{n \to \infty} \int_a^b x^2 \, dx = \frac{b^3}{3} - \frac{a^3}{3}. \]
Final Answer: \[ \boxed{\frac{b^3}{3} - \frac{a^3}{3}}. \]