Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(\int_{0}^{1} f(x) d x=2, \int_{1}^{2} f(x) d x=3, \int_{0}^{1} g(x) d x=-1\), and \(\int_{0}^{2} g(x) d x=4\). Use properties of definite integrals (linearity, interval additivity, and so on) to calculate each of the integrals in Problems 9-16. $$ \int_{1}^{1}[3 f(x)+2 g(x)] d x $$

Step-by-Step Solution

Answer

The integral is 0 due to the zero-length interval property.

Step 1: Understand the Integral Limits

The integral given is \[ \int_{1}^{1}[3 f(x) + 2 g(x)] dx.\] At first glance, notice that the upper and lower limits of the integral are the same (both are 1). This setup is crucial for solving the problem, because there is a fundamental property of definite integrals which we need to apply.

Step 2: Apply the Zero-Length Interval Property

Recall the property that if the upper and lower limits of a definite integral are the same, then the value of the integral is 0. Formally stated, for any function \( h(x) \) and any point \( a \), \[\int_{a}^{a} h(x) \, dx = 0.\] Therefore, applying this property directly to our problem: \[\int_{1}^{1}[3 f(x) + 2 g(x)] dx = 0.\] No matter what's inside the brackets, the evaluation is 0 because the bounds are the same.

Key Concepts

Linearity of Integration

Linearity of integration is a fundamental property of definite integrals, which states that the integral of a sum of functions is the sum of their integrals. This property also allows us to factor out constants from the integral sign. In mathematical terms, this means:

• For any two functions, say \( f(x) \) and \( g(x) \), and constants \( a \) and \( b \), the integral of their linear combination is given by:

\[ \int (a \, f(x) + b \, g(x)) \, dx = a \int f(x) \, dx + b \int g(x) \, dx \]This property simplifies the process of evaluating integrals by breaking them down into manageable parts.
By taking each part of the combination separately, we can easily handle complex integrals. It is a very handy tool for mathematicians and is fundamental in calculus.
Understanding linearity is essential because it allows integration to maintain its "additive" nature, keeping calculations straightforward, especially when paired with other integration properties.

Properties of Definite Integrals

Definite integrals come with a bundle of useful properties that make computations easier and are valuable when solving real-world problems. Here are some key properties:

• Linearity: As mentioned, it allows for the splitting of the integral of a sum into the sum of integrals, as well as factorization of constants.
• Interval Additivity: This states that if a function is integrated over two adjacent intervals, the result is the sum of the integrals over each interval:

\[ \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx = \int_{a}^{c} f(x) \, dx \]

• Symmetry: If a function is symmetric about the y-axis, the integral over a symmetric interval centered on the origin may simplify.

These properties not only simplify calculations but also enhance our understanding of integral behavior. They reveal integral calculus as not just a mechanical tool, but a field rich with insights into functions and their transformations.

Zero-Length Interval Property

The zero-length interval property of definite integrals is a fascinating and very intuitive concept. It states that if the upper and lower limits of an integral are the same, the integral evaluates to zero:

• Symbolically, for any function \( h(x) \) and any point \( a \):

\[ \int_{a}^{a} h(x) \, dx = 0 \]This property makes sense if you think about the integral as measuring the 'area' under a curve from one point to another. If both points are the same, essentially no area is captured, leading to a result of zero.
In practical applications, this property allows us to instantly recognize and solve integrals without needing to perform any detailed calculations when the limits match. It's a key example of how sometimes the absence of movement (along an interval) results in zero, highlighting a fundamental aspect of calculus as it connects to the physical world.