Calculating Area Under a Piecewise Function

The area under the curve of a function defined over an interval is calculated by summing the areas of the sections defined by the function's definition. For a piecewise constant function, this area is the sum of the areas of rectangles, where each rectangle's area is its width multiplied by its height.

Area Calculation Steps

The function $y(x)$ is defined over the interval $[0, 1]$ in three parts:

• Interval 1: $0 \le x < \frac{1}{3}$
  • Function value (height): $y(x) = 2$
  • Interval width: $\Delta x_1 = \frac{1}{3} - 0 = \frac{1}{3}$
  • Area 1: $A_1 = \text{height} \times \text{width} = 2 \times \frac{1}{3} = \frac{2}{3}$
• Interval 2: $\frac{1}{3} \le x < \frac{3}{4}$
  • Function value (height): $y(x) = 3$
  • Interval width: $\Delta x_2 = \frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}$
  • Area 2: $A_2 = \text{height} \times \text{width} = 3 \times \frac{5}{12} = \frac{15}{12} = \frac{5}{4}$
• Interval 3: $\frac{3}{4} \le x \le 1$
  • Function value (height): $y(x) = 1$
  • Interval width: $\Delta x_3 = 1 - \frac{3}{4} = \frac{1}{4}$
  • Area 3: $A_3 = \text{height} \times \text{width} = 1 \times \frac{1}{4} = \frac{1}{4}$

Total Area

The total area under the curve is the sum of the areas from the three intervals:

$ \text{Total Area} = A_1 + A_2 + A_3 $

$ \text{Total Area} = \frac{2}{3} + \frac{5}{4} + \frac{1}{4} $

First, add the fractions with the same denominator:

$ \frac{5}{4} + \frac{1}{4} = \frac{6}{4} = \frac{3}{2} $

Now, add this result to the area from the first interval:

$ \text{Total Area} = \frac{2}{3} + \frac{3}{2} $

Find a common denominator, which is 6:

$ \text{Total Area} = \frac{2 \times 2}{3 \times 2} + \frac{3 \times 3}{2 \times 3} = \frac{4}{6} + \frac{9}{6} $

$ \text{Total Area} = \frac{4 + 9}{6} = \frac{13}{6} $

The area under the curve for the interval $[0, 1]$ is $\frac{13}{6}$.