Question

Find the area of the region bounded by the curve \( y = x^2 \), and the lines \( x = 1 \), \( x = 2 \), and \( y = 0 \).

Hint:

To calculate areas under curves, use the formula \( \int_{a}^{b} f(x) \,dx \), where \( f(x) \) is the function representing the curve.

Answer 1

Step 1: Apply Definite Integration for Area
The area under the curve is given by the integral: \[ A = \int_{1}^{2} x^2 \,dx. \] Step 2: Compute the Integral
Evaluating the integral, we get: \[ A = \left[ \frac{x^3}{3} \right]_{1}^{2}. \] \[ = \frac{2^3}{3} - \frac{1^3}{3}. \] \[ = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}. \]