Problem 18 - The Calculus with Analytic Geometry

Question

Show that if \(f\) is continuous on \([-1,2]\), then $$ \int_{-1}^{2} f(x) d x+\int_{2}^{0} f(x) d x+\int_{0}^{1} f(x) d x+\int_{1}^{-1} f(x) d x=0 $$

Step-by-Step Solution

Answer

Summing those integrals results in zero as they cancel each other out by covering the same path twice in opposite directions.

Step 1: Review the integrals

Observe the integrals given: - - The bounds on each integral include the whole interval from - - - -

Step 2: Reverse integration property

Using the property of integrals where reversing the order of integration negates the integral: - - - -

Step 3: Combine the integrals

Combine integrals based on their limits using the property of linearity of integrals: So combining the integrals gives:

Step 4: Conclude

Since results in the same path covered twice in opposite directions, it sums to zero:

Key Concepts

Continuous Functions

A continuous function is one where small changes in the input result in small changes in the output. In other words, there are no sudden jumps or breaks in the graph of the function. For example, the function\(f(x)\) is continuous on the interval \([-1,2]\). This means that for every point \(-1 \leq x \leq 2\), the function value changes smoothly.

Definite Integrals

Definite integrals are a way to calculate the area under a curve between two limits, here called bounds. The definite integral of a function \(f(x)\) from \(a\) to \(b\) is written as \(\int_{a}^{b} f(x) dx\). It gives us the net area between \(f(x)\) and the x-axis.
For example, in the given exercise, the integrals \(\int_{-1}^{2} f(x) dx\), \(\int_{2}^{0} f(x) dx\), and so forth represent different parts of the interval from \(-1\) to 2. Understanding how to compute these integrals is crucial for solving this problem.

Properties of Integrals

Integrals have several key properties that make them easier to handle. Here are some important ones:

• Linearity: If \(f(x)\) and \(g(x)\) are integrable functions and \(c\) is a constant, then \(\int (cf(x) + g(x)) dx = c \int f(x) dx + \int g(x) dx \).


• Additivity: If \(a\le c \le b\), then\( \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx\).


• Reversing limits: For any continuous function \(f(x)\), reversing the limits of integration introduces a negative sign, i.e., \(\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx\).
  
  An essential property we use in the exercise is the reversal of limits to combine and simplify the given integrals.
  

Linearity of Integrals

The property of linearity is fundamental when working with integrals. It tells us that we can separate and combine integrals involving sums and constants.
For instance, given the exercise, we observe:
\(\int_{-1}^{2} f(x) dx + \int_{2}^{0} f(x) dx + \int_{0}^{1} f(x) dx + \int_{1}^{-1} f(x) dx\). Applying linearity, and by reordering and combining integrals, we see that integrals over the same intervals but in reversed orders cancel each other out.

Order of Integration

The order of integration is important in solving integrals. Specifically, changing the integration limits can help in simplifying complex expressions.
For example, in the exercise, \(\int_{2}^{0} f(x) dx = -\int_{0}^{2} f(x) dx \). This helps us rearrange and combine integrals. Thus, understanding the order of integration can simplify otherwise complex calculations.