Question

True or False? In Exercises 55-60, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{a}^{b}[f(x)+g(x)] d x=\int_{a}^{b} f(x) d x+\int_{a}^{b} g(x) d x $$

Short answer

The statement is True.

Step-by-step solution

Checking the Rule

The rule that is being examined in the exercise is about integration of a sum of a function, specifically, \(\int_{a}^{b}[f(x)+g(x)] d x=\int_{a}^{b} f(x) d x+\int_{a}^{b} g(x) d x\). This rule is a well-known property of definite integrals.

True or False?

The given statement is true since one of the properties of definite integrals is linearity, which states that the integral of the sum of multiple functions is equal to the sum of the integrals of these functions. Thus, the integral of \(f(x) + g(x)\) from \(a\) to \(b\) is indeed equal to the integral of \(f(x)\) from \(a\) to \(b\) plus the integral of \(g(x)\) from \(a\) to \(b\)

Key concepts

Linearity of Integrals

Understanding the linearity of integrals is crucial in calculus as it simplifies the process of integrating complex functions. When we talk about the linearity of integrals, we refer to two primary things - the integral of a constant multiplied by a function, and the integral of the sum of two functions.

Let's take the first one. If you have a constant, say, 'c', and a function, 'f(x)', the linearity property tells us that \( \int_{a}^{b}cf(x)dx = c\int_{a}^{b}f(x)dx \). This means you can pull the constant outside the integral and then integrate the function normally. It's like saying, 'First, let's deal with the constant, and then we'll focus on the function.' This property is immensely useful when dealing with coefficients in integrals.

The second part of linearity involves adding functions together – precisely what our textbook exercise assesses. If you're given two functions, 'f(x)' and 'g(x)', and you need to integrate their sum over the same interval, you are allowed to split the integral into two, handling each function individually - this is very similar to how you would handle addition in basic arithmetic. In formula terms this reads as \( \int_{a}^{b} [f(x) + g(x)]dx = \int_{a}^{b} f(x)dx + \int_{a}^{b} g(x)dx \). So, if you ever get a compound function like this, remember you can break it down and simplify your life!

Integral of a Sum

Dealing with the integral of a sum can seem daunting at first, but with our understanding of the linearity of integrals, it's just a piece of cake. When you come across an expression such as \( \int_{a}^{b}[f(x) + g(x)]dx \), it's nothing but an application of the linearity property.

In practical terms, you're basically taking a sum inside an integral and breaking it into two separate integrals of each function, over the same interval. It's like having a basket of apples and oranges and deciding to count them separately rather than together. By applying this approach, you individually integrate 'f(x)' and 'g(x)' over the interval from 'a' to 'b'.

This process not only simplifies calculations but also allows for easier application of other integration techniques, should either 'f(x)' or 'g(x)' require them. It's important to note that this property holds true only for finite sums of functions. Once you start exploring infinite series, integration becomes a bit trickier and requires additional rules and tools.

True or False Exercises in Calculus

True or false exercises in calculus, like the one from our textbook, are instrumental in reinforcing important concepts and testing a student's understanding. Such exercises typically present a mathematical statement and ask you to judge its validity. It's not enough to know the answer; being able to explain it is where the real learning happens.

In calculus, these exercises often involve common rules and properties, like the linearity of integrals or the fundamental theorem of calculus. They could be tricky, presenting statements that seem plausible but are false or vice versa. When tackling these exercises, it's best to consider what you know about the functions and the properties involved, and not to hesitate using a simple example to test the given statement.

In the case of our textbook problem, the statement about the integral of a sum is true, which aligns with the linearity property of integrals. Encountering true or false questions helps to solidify these concepts, making it easier to recognize and apply the properties correctly in more complex problems.