Question

Sketch the graph of \( y = x|x| \) and hence find the area bounded by this curve, the X-axis, and the ordinates \( x = -2 \) and \( x = 2 \), using integration.

Hint:

When finding the area bounded by curves, split the integral into regions where the function behaves differently (e.g., absolute values or piecewise functions).

Answer 1

Step 1: Define the function \( y = x|x| \)
The function \( y = x|x| \) can be written piecewise as: \[ y = \begin{cases} -x^2, & x < 0 \\ x^2, & x \geq 0 \end{cases} \]
Step 2: Visualize the function's graph
The graph of \( y = x|x| \) resembles a parabola. It is concave downwards for \( x<0 \) and concave upwards for \( x \geq 0 \). (See attached graph.) 

Step 3: Set up the area calculation using integration
The area of the shaded region from \( x = -2 \) to \( x = 2 \) is calculated as: \[ \text{Area} = \int_{-2}^{2} |y| \, dx = 2 \int_{0}^{2} x^2 \, dx. \] 
Step 4: Compute the integral
\[ \int_{0}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{2} = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}. \] Therefore, the total area is: \[ \text{Area} = 2 \times \frac{8}{3} = \frac{16}{3}. \] Step 5: State the final answer
The area enclosed by the curve \( y = x|x| \), the X-axis, and the vertical lines \( x = -2 \) and \( x = 2 \) is: \[ \boxed{\frac{16}{3}}. \]