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Chapter 5: Problem 8

\(\quad a\) and \(b\) are constants, and \(x\) and \(t\) are variables. In these activities, identify each notation as always representing a function of \(x,\) a function of \(t,\) or a number. a. \(\int f(t) d t\) b. \(\int f(x) d x\) c. \(\int_{a}^{b} f(t) d t\)

Short Answer

Expert verified
a. Function of \(t\); b. Function of \(x\); c. Number.

Step by step solution

01

Analyze the Notation

In each part of the exercise, identify the function and the variable of integration. The notation \( \int f(t) dt\) suggests that the function \(f(t)\) is defined in terms of the variable \(t\), and the integration is performed with respect to \(t\). Similarly, \(\int f(x) dx\) suggests that \(f(x)\) is a function of \(x\), integrated with respect to \(x\).
02

Determine Function of t

For part (a), \(\int f(t) dt\), the expression represents an indefinite integral of the function of \(t\). Since the integral is with respect to \(t\), \(\int f(t) dt\) represents a function of \(t\).
03

Determine Function of x

For part (b), \(\int f(x) dx\), the expression represents an indefinite integral of the function of \(x\). Since the integral is with respect to \(x\), \(\int f(x) dx\) represents a function of \(x\).
04

Determine the Result of Definite Integral

For part (c), \(\int_{a}^{b} f(t) dt\), this is a definite integral evaluated from \(a\) to \(b\), both of which are constants. The result of this definite integral is a number, as it gives the net area under the curve \(f(t)\) from \(t = a\) to \(t = b\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Indefinite Integral
An indefinite integral, symbolized as \( \int f(x) dx \), represents a family of functions whose derivative is \( f(x) \). In this context, the integral lacks specific limits, meaning it does not evaluate to a set number. Rather, it includes an arbitrary constant known as \( C \). This constant accounts for any possible vertical shifts of the antiderivative on the graph.

Key Facts:
  • The indefinite integral of \( f(x) \) with respect to \( x \) is often written as \( F(x) + C \).
  • \( \frac{d}{dx}(F(x) + C) = f(x) \), which confirms \( F(x) \) is an antiderivative.
  • The indefinite integral is useful in problems where we seek a general solution to differential equations.
When you see \( \int f(t) dt \), you're looking at an indefinite integral with respect to \( t \). Essentially, you are determining a function that, when differentiated with respect to \( t \), gives \( f(t) \).
Definite Integral
A definite integral differs from an indefinite integral because it evaluates the net area under the curve of a function \( f(t) \) between two constants \( a \) and \( b \). It is represented as \( \int_{a}^{b} f(t) dt \).

Here鈥檚 what happens during a definite integration process:
  • The function \( f(t) \) is integrated with respect to \( t \) and yields an antiderivative \( F(t) \).
  • The antiderivative is then evaluated at the upper limit \( b \) and the lower limit \( a \).
  • The result is \( F(b) - F(a) \), providing a real number that can represent the net area.
If \( \int_{a}^{b} f(t) dt = F(b) - F(a) \), it describes the accumulation of quantities or the total change, and it becomes highly valuable in applications like physics for calculating work done or probability distributions in statistics.
Variables of Integration
The variable of integration helps determine the nature of the antiderivative and the limits of the integration. When an integral is expressed, such as \( \int f(x) dx \), \( x \) is the variable of integration.

Key Roles of the Variable of Integration:
  • It signifies the variable according to which the function is being integrated.
  • Adjusting the integration variable alters the problem statement, meaning \( \int f(t) dt \) differs from \( \int f(x) dx \), even though the functions might initially look similar.
  • In definite integrals \( \int_{a}^{b} f(t) dt \), it shows the specific intervals over interpretation is made.
Remember, this variable dictates how you differentiate your function to verify your integral results. It acts as a placeholder during integration, fundamentally affecting how expressions are evaluated in both indefinite and definite integrals.

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Most popular questions from this chapter

High Dive According to the Guinness Book of Records, the world's record high dive from a diving board is 176 feet, 10 inches. It was made by Olivier Favre (Switzerland) in \(1987 .\) Ignoring air resistance, approximate Favre's impact velocity (in miles per hour) from a height of 176 feet, 10 inches.

Swim Time The rate of change of the winning times for the men's 100 -meter butterfly swimming competition at selected summer Olympic games between 1956 and 2000 can be modeled as \(w(t)=0.0106 t-1.148\) seconds per year where \(t\) is the number of years since \(1900 .\) (Source: Based on data from Swim World) a. Calculate the average rate of change of the winning times for the competition from 1956 through \(2000 .\) b. Illustrate the average rate of change of swim time on a graph of \(w\) c. Illustrate the average rate of change of swim time on a graph of the particular antiderivative of \(w\) where \(W(0)=0 .\)

Phone Calls The rate of change of the number of international telephone calls billed in the United States between 1980 and 2000 can be described by \(P(x)=32.432 e^{0.1826 x}\) million calls per year where \(x\) is the number of years since \(1980 .\) (Source: Based on data from the Federal Communications Commission) a. Evaluate \(\int_{5}^{15} p(x) d x\) b. Interpret the answer from part \(a\)

For Activities 1 through \(10,\) write the general antiderivative. $$ \int(5.6 \cos x-3) d x $$

\(\quad a\) and \(b\) are constants, and \(x\) and \(t\) are variables. In these activities, identify each notation as always representing a function of \(x,\) a function of \(t,\) or a number. a. \(\frac{d}{d x} \int_{a}^{x} f(t) d t\) b. \(\frac{d}{d t} \int_{a}^{t} f(x) d x\) c. \(\frac{d}{d x} \int_{a}^{a} f(t) d t\)
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