Problem 92 - Calculus Volume 1

Question

Suppose that \(\int_{0}^{4} f(x) d x=5\) and \(\int_{0}^{2} f(x) d x=-3,\) and \(\int_{0}^{4} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=2 .\) In the following exercises, compute the integrals. \(\int_{0}^{2}(3 f(x)-4 g(x)) d x\)

Step-by-Step Solution

Answer

The integral \( \int_{0}^{2}(3 f(x)-4 g(x)) dx = -17.\)

Step 1: Break Down the Integral

The integral given is \( \int_{0}^{2}(3f(x) - 4g(x)) \ dx \). Since integrals are linear, we can separate the terms and handle them individually: \[ \int_{0}^{2} 3f(x) \ dx - \int_{0}^{2} 4g(x) \ dx. \] This separation is allowed because integration distributes over addition and subtraction.

Step 2: Factor Out Constants

You can factor out constants from integrals. Therefore, we rewrite the separated terms as: \[ 3 \int_{0}^{2} f(x) \ dx - 4 \int_{0}^{2} g(x) \ dx. \] This allows us to use the given values of the integrals in the problem.

Step 3: Substitute Known Integral Values

Now substitute the known integral values into the equation: \( \int_{0}^{2} f(x) \ dx = -3 \) and \( \int_{0}^{2} g(x) \ dx = 2 \). So, the equation becomes: \[ 3(-3) - 4(2). \]

Step 4: Perform Arithmetic Calculations

Calculate the values separately: \[ 3(-3) = -9 \text{ and } 4(2) = 8. \] Combine these results to find the final answer: \[ -9 - 8 = -17. \]

Key Concepts

Definite Integral

A definite integral is a mathematical concept used to calculate the area under a curve from a specific start point to an end point. In terms of calculus, it provides the accumulated quantity, such as area or distance, between two bounds on a particular function. The result of a definite integral is a number rather than a function. For example, given the function \( f(x) \), the definite integral from \( a \) to \( b \) is expressed as \( \int_{a}^{b} f(x) \, dx \). This calculation accounts for all the infinitesimal areas beneath the curve of \( f(x) \) within that interval.

Here are some key features of definite integrals:

• The limits of integration \( a \) and \( b \) determine the interval of integration.
• The result is a fixed number, representing the net area between the function and the x-axis.
• Definite integrals can be broken down or combined due to their linear properties, allowing for expressions like \( \int_{a}^{b} [3f(x) - 4g(x)] \, dx \).

Learning about definite integrals is essential for understanding areas, motion, and further calculus applications.

Discrete Mathematics

Discrete mathematics deals with distinct and separate values, such as integers, graphs, and logical statements, rather than continuous ranges. While it may seem unrelated to calculus, it plays a vital role in understanding integral concepts through logical reasoning and computation techniques. Topics in discrete mathematics include set theory, combinatorics, and logic, all of which require precise calculation, much like calculating integrals.

In the context of integrals, discrete mathematics contributes by:

• Providing a foundation in mathematical reasoning for interpreting and solving integrals.
• Facilitating the computation of summations and sequences that can approximate integrals.

Understanding discrete mathematics aids in developing a robust logical framework necessary for tackling calculus problems, including integration.

Mathematical Reasoning

Mathematical reasoning is the logical thought process used to analyze and solve problems. It is crucial in calculus for verifying solutions, such as those involving integrals, and understanding why certain properties hold. In the context of the given exercise, mathematical reasoning helps determine why integrals can be separated and constants factored out.

Here’s how mathematical reasoning applies to integrals:

• Linearity: Understanding that integrals are linear helps justify the separation and factoring of constants in the problem \( \int_{0}^{2}(3f(x) - 4g(x)) \, dx \).
• Substitution of known values: Mathematically reasoning the use of previously computed integral values helps streamline calculations and validate results.

By honing mathematical reasoning skills, one can better understand, interpret, and solve integrals and other calculus-related problems.