Question

Evaluate \( \int_{0}^{2} x^2 \, dx \) and hence show the region on the graph whose area it represents.

Hint:

The definite integral \( \int_a^b f(x) \, dx \) represents the area under the curve \( y = f(x) \) between \( x = a \) and \( x = b \).

Answer 1

Step 1: Use the formula for definite integrals: \[ \int x^n dx = \frac{x^{n+1}}{n+1} + C. \] 
Step 2: Compute the integral: \[ \int_{0}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{2} = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3} - 0 = \frac{8}{3}. \] Thus, the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) is \( \frac{8}{3} \).