Question

89-90. Integrating piecewise continuous functions Use geometry and the result of Exercise 88 to evaluate the following integrals. $$\int_{0}^{10} f(x) d x, \text { where } f(x)=\left\\{\begin{array}{ll} 2 & \text { if } 0 \leq x \leq 5 \\ 3 & \text { if } 5

Short answer

Question: Evaluate the definite integral of the following piecewise continuous function from 0 to 10: \(f(x) = 2\) if \(0 \leq x \leq 5\) and \(f(x) = 3\) if \(5 < x \leq 10\). Answer: The definite integral of the given piecewise continuous function from 0 to 10 is 25.

Step-by-step solution

Split the integral into two parts

Since the function changes its value at \(x=5\), we need to split the integral into two parts: one that covers the interval from \(0\) to \(5\) and another that covers the interval from \(5\) to \(10\). The integral can be rewritten as: $$\int_{0}^{10} f(x) d x = \int_{0}^{5} f(x) d x +\int_{5}^{10} f(x) d x$$

Use geometric interpretation to find the integral values for each part

In the first interval, \(0 \leq x \leq 5\), the function takes the constant value \(2\), which means we have a rectangle with width 5 and height 2. Similarly, in the second interval, \(5 < x \leq 10\), the function takes the constant value \(3\), which means we have a rectangle with width 5 and height 3. To find the area under the curve (integral), we need to find the area of each rectangle and add them together.

Find the area of the rectangles for the two intervals

The area of the rectangle in the first interval is: $$\text{Area}_1 = 2 \times 5 = 10$$ The area of the rectangle in the second interval is: $$\text{Area}_2 = 3 \times 5 = 15$$

Add the two areas to get the value of the integral

The value of the integral now is given by the sum of the areas of the two rectangles: $$\int_{0}^{10} f(x) d x = \text{Area}_1 + \text{Area}_2 = 10 + 15 = 25$$ Therefore, the value of the integral is 25.

Key concepts

Integral Calculus

Integral calculus is a fundamental part of mathematics that deals with the concept of integration, which is essentially the reverse process of differentiation. It allows us to determine various quantities such as areas under curves, volumes, and other physical properties by aggregating infinitesimal data points. When solving integrals, especially for piecewise functions, the strategy is to decompose the integral into manageable segments.

For the given piecewise function, we first identify where the function changes. This involves recognizing the different parts of the function that have distinct behaviors. Each part of the function is treated separately, and we calculate the integral for each section. Once obtained, we sum these integral values to find the total area under the curve from start to finish. This technique leverages the linearity property of integrals, allowing us to break down complex problems into simpler, solvable pieces.

Piecewise Continuous Functions

Piecewise continuous functions are functions defined by multiple sub-functions, each applying to a specific interval of the domain. They are very useful in modeling situations where a function might change behavior abruptly, such as different cost rates over time or varying speeds of a vehicle on different road segments.

In the context of integration, a piecewise continuous function requires us to identify the intervals where each sub-function holds. This is why a critical first step is splitting the integral at the points of discontinuity or where the function's definition changes. For example, in this exercise, the function is defined as 2 for the interval [0, 5] and changes to 3 for (5, 10]. By understanding this, we apply the respective constant value over each interval to evaluate the integral precisely. This approach ensures we can handle complex real-world functions with discontinuities while still managing to perform integration effectively.

Geometric Interpretation of Integrals

The geometric interpretation of integrals provides an intuitive grasp of what integration accomplishes. Essentially, integration calculates the 'area under the curve' of a graph of a function—shaded areas that give us tangible insight into the accumulation of quantities.

For piecewise functions, this concept translates to finding the areas of simple geometric shapes like rectangles. In this exercise, the given function yields two rectangles distinguished by its constant value on each segment. The area of each rectangle can be easily found using the formula for the area: width multiplied by height. By adding these areas together, we obtain the total integral of the piecewise function over its entire domain.

Such a visual and geometric approach aids in comprehending the functionality and outcome of integrals, particularly in situations involving constant functions or linear combinations. This understanding not only simplifies the calculation but also enhances our conceptual appreciation of integral calculus.