I - Integration Module Overview

Integration Module Overview

Introduction

Integration is often thought of as the reverse process of differentiation or as reconstructing a function from its derivative. As you will discover, the derivative and the integral are closely related by the Fundamental Theorem of Calculus. It is imperative to understand the similarities and differences in the meanings of indefinite integral, antiderivative, definite integral, and limit of an approximating sum. Of particular interest and applicability are the interpretation of a definite integral as an area and the usefulness of antidifferentiation in solving initial-value problems and analyzing rectilinear motion.

Essential Questions

What is an antiderivative?
How are integrals related to derivatives?
When must Riemann sums be used instead of integration?
What is the relationship between an integral and area?
How can integration properties be used to simplify integration?
How is the Fundamental Theorem of Calculus used to evaluate definite integrals?
What is meant by substitution of variables and change of limits for definite integrals?
What is the net change of the quantity over the interval?
What is the average value of a function along a given interval?
How do you find the distance traveled by a particle along a line?
How can numerical techniques be applied to compute an integral without knowing the associated antiderivative?
How do you determine accumulated change from a rate of change?
What features of graphing technology support definite integration?

Key Terms

The following key terms will help you understand the content in this module.

Accumulated change - Based on using the integral of a rate of change function f(t) to give the amount of change over the interval a ≤ t ≤ b.

Antiderivative - A function F(x) whose derivative f(x) is given by the function F'(x) = f(x) on an interval I for all x in I.

Area of a plane region - Area bounded by a curve y = f(x) on a closed interval [a, b] and one or more additional curves.

Average value of a function - If f is integrable on the closed interval [a,b], then $LaTeX: f_{avg}=\frac{1}{b-a}\int_a^{b}f\left(x\right)dx$ $f_{avg}=\frac{1}{b-a}\int_a^{b}f\left(x\right)dx$ .

Constant of integration - An arbitrary constant representing the vertical translation of the graph of an antiderivative F of f.

Definite integral - A number that is the limit of the Riemann sums $LaTeX: \lim_{||\Delta ||\to0}=\sum_{i=1}^{n}=f(c_{i}) \Lambda x_{i}$ $\lim_{||\Delta ||\to0}=\sum_{i=1}^{n}=f(c_{i}) \Lambda x_{i}$ for a function f that is defined on the closed interval [a,b] and is denoted as $LaTeX: \int_a^{b}f\left(x\right)dx$ $\int_a^{b}f\left(x\right)dx$ .

Differential equation - An equation that involves a derivative of a function.

fnInt - A calculator calculus command that computes an approximation to the definite integral of f with respect to a given variable from a to b . Command syntax is fnInt(f (variable), variable, a,b) .

Fundamental Theorem of Calculus - A theorem stating the relationship between integration and differentiation. If f(x) is continuous on [a,b], then $LaTeX: \int_a^{b}f\left(x\right)dx$ $\int_a^{b}f\left(x\right)dx$ , where F is any antiderivative of f, i.e., a function such that F'= f.

Indefinite integral - The set of all antiderivatives of a function f(x) that is symbolized by $LaTeX: \int f\left(x\right)dx$ $\int f\left(x\right)dx$ .

Net change - The integral of a function\'s rate of change over an interval a ≤ x ≤ b, without regard to possible sign change of the function on [a,b].

Numerical integration - The approximate computation of an integral (area under a curve) using numerical techniques used when it is not possible to find a formula that can be evaluated to give the value of a definite integral.

Rectilinear motion - Motion of an object moving in a straight line.

Riemann sum - Any sum of the form , $LaTeX: \sum_{i=1}^{n}=f(c_{i}) \Lambda x_{i}$ $\sum_{i=1}^{n}=f(c_{i}) \Lambda x_{i}$ where $LaTeX: \Delta x_i$ $\Delta x_i$ is the width of the ith subinterval for any partition with a=x₀<x₁<x₂<. . . $LaTeX: a=x_0<x_1<x_2<...<x_{n-1}<x_n=b\:where\:x_{i-1}<c_1<x_i$ $a=x_0<x_1<x_2<...<x_{n-1}<x_n=b\:where\:x_{i-1}<c_1<x_i$ .

Second Fundamental Theorem of Calculus - If f(x) is continuous on an open interval I containing a, then for every x in the interval, $LaTeX: \frac{d}{dx}\left[\int_a^{b}f\left(t\right)dt\right]=f\left(x\right)$ $\frac{d}{dx}\left[\int_a^{b}f\left(t\right)dt\right]=f\left(x\right)$ .

Sigma notation - The Greek capital letter sigma 𝚺 used to indicate summation and often written $LaTeX: \sum_{i=a}^{b}=f(c_{i}) \Lambda x_{i}=x_a+x_{a+1}+...+x_b$ $\sum_{i=a}^{b}=f(c_{i}) \Lambda x_{i}=x_a+x_{a+1}+...+x_b$ for the sum of the elements LaTeX: x_i $x_i$ from i = a to b.

Upper and lower sums - The sum of the areas of rectangles subdividing the interval [a,b] into n subintervals, each of width $LaTeX: \Delta x=\frac{\left(b-a\right)}{n}$ $\Delta x=\frac{\left(b-a\right)}{n}$ The upper sum is the area of circumscribed rectangles denoted by $LaTeX: S\left(n\right)=\sum_{i=1}^{n}=f(M_{i}) \Lambda x$ $S\left(n\right)=\sum_{i=1}^{n}=f(M_{i}) \Lambda x$ . The lower sum is the area of inscribed rectangles denoted by $LaTeX: s\left(n\right)=\sum_{i=1}^{n}=f(m_{i}) \Lambda x$ $s\left(n\right)=\sum_{i=1}^{n}=f(m_{i}) \Lambda x$ .

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