Home The Six Pillars of CalculusThe Pillars: A Road MapA picture is worth 1000 words Trigonometry ReviewThe basic trig functionsBasic trig identities The unit circle Addition of angles, double and half angle formulas The law of sines and the law of cosines Graphs of Trig Functions Exponential FunctionsExponentials with positive integer exponentsFractional and negative powers The function $f(x)=a^x$ and its graph Exponential growth and decay Logarithms and Inverse functionsInverse FunctionsHow to find a formula for an inverse function Logarithms as Inverse Exponentials Inverse Trig Functions Intro to LimitsOverviewDefinition Onesided Limits When limits don't exist Infinite Limits Summary Limit Laws and ComputationsLimit LawsIntuitive idea of why these laws work Two limit theorems How to algebraically manipulate a 0/0? Indeterminate forms involving fractions Limits with Absolute Values Limits involving indeterminate forms with square roots Limits of Piecewise Functions The Squeeze Theorem Continuity and the Intermediate Value TheoremDefinition of continuityContinuity and piecewise functions Continuity properties Types of discontinuities The Intermediate Value Theorem Summary of using continuity to evaluate limits Limits at InfinityLimits at infinity and horizontal asymptotesLimits at infinity of rational functions Which functions grow the fastest? Vertical asymptotes (Redux) Summary and selected graphs Rates of ChangeAverage velocityInstantaneous velocity Computing an instantaneous rate of change of any function The equation of a tangent line The Derivative of a Function at a Point The Derivative FunctionThe derivative functionSketching the graph of $f'$ Differentiability Notation and higherorder derivatives Basic Differentiation RulesThe Power Rule and other basic rulesThe derivative of $e^x$ Product and Quotient RulesThe Product RuleThe Quotient Rule Derivatives of Trig FunctionsNecessary LimitsDerivatives of Sine and Cosine Derivatives of Tangent, Cotangent, Secant, and Cosecant Summary The Chain RuleTwo Forms of the Chain RuleVersion 1 Version 2 Why does it work? A hybrid chain rule Implicit DifferentiationIntroductionExamples Derivatives of Inverse Trigs via Implicit Differentiation A Summary Derivatives of LogsFormulas and ExamplesLogarithmic Differentiation Derivatives in ScienceIn PhysicsIn Economics In Biology Related RatesOverviewHow to tackle the problems Example (ladder) Example (shadow) Linear Approximation and DifferentialsOverviewExamples An example with negative $dx$ Differentiation ReviewHow to take derivativesBasic Building Blocks Advanced Building Blocks Product and Quotient Rules The Chain Rule Combining Rules Implicit Differentiation Logarithmic Differentiation Conclusions and Tidbits Absolute and Local ExtremaDefinitionsThe Extreme Value Theorem Critical Numbers Steps to Find Absolute Extrema The Mean Value and other TheoremsRolle's TheoremsThe Mean Value Theorem Finding $c$ $f$ vs. $f'$Increasing/Decreasing Test and Critical NumbersProcess for finding intervals of increase/decrease The First Derivative Test Concavity Concavity, Points of Inflection, and the Second Derivative Test The Second Derivative Test Visual Wrapup Indeterminate Forms and L'Hospital's RuleWhat does $\frac{0}{0}$ equal?Examples Indeterminate Differences Indeterminate Powers Three Versions of L'Hospital's Rule Proofs OptimizationStrategiesAnother Example Newton's MethodThe Idea of Newton's MethodAn Example Solving Transcendental Equations When NM doesn't work AntiderivativesAntiderivativesCommon antiderivatives Initial value problems Antiderivatives are not Integrals The Area under a curveThe Area Problem and ExamplesRiemann Sum Notation Summary Definite IntegralsDefinition of the IntegralProperties of Definite Integrals What is integration good for? More Applications of Integrals The Fundamental Theorem of CalculusThree Different ConceptsThe Fundamental Theorem of Calculus (Part 2) The Fundamental Theorem of Calculus (Part 1) More FTC 1 The Indefinite Integral and the Net ChangeIndefinite Integrals and AntiderivativesA Table of Common Antiderivatives The Net Change Theorem The NCT and Public Policy SubstitutionSubstitution for Indefinite IntegralsExamples to Try Revised Table of Integrals Substitution for Definite Integrals Examples Area Between CurvesComputation Using IntegrationTo Compute a Bulk Quantity The Area Between Two Curves Horizontal Slicing Summary VolumesSlicing and Dicing SolidsSolids of Revolution 1: Disks Solids of Revolution 2: Washers More Practice 
What is integration good for?As we have seen, definite integrals can be used to compute area, just as derivatives can be used to compute slopes of tangent lines. But there is much more to derivatives than slopes of tangent lines, and there is much more to integrals than area. Integrals can be used to compute any bulk quantity.
Example: DistanceSuppose that we know a particle's velocity as a function of time.
If the velocity is constant, then it's easy to find the distance
traveled: $$ \hbox{Rate} \times \hbox{Time} = \hbox{Constant
Velocity} \times \hbox{Time} = \hbox{Distance}.$$ If the velocity
is almost constant, then we
can get a good approximation by pretending that it is constant. If
the velocity varies a lot, then we break our time interval into
little subintervals on which the velocity doesn't change much,
estimate each one, and add up the pieces. This is exactly the same
strategy that we used for area under a curve. Example 1: Find the distance traveled by a person walking
at 2 MPH from 1:00 PM to 4:00 PM. The answer in Example 2 is an underestimate, a little bit too small, since the velocity goes up from $v=2.02$ at $t=1$ to $v=2.08$ at $t=4$. If we wanted an overestimate of the area, we would use the value of $v(t)$ at the right endpoint instead: $(41)v(4)=3(2.08)=6.24$. And if we wanted a more accurate guess, we might use the midpoint $f(2.5)=2.05$ and estimate the area as $(41)v(2.5)=3(2.05)=6.015$. But whether we use the left endpoint, the right endpoint, or the midpoint, we get roughly the same answer, since the velocity just isn't changing very much between $t=1$ and $t=4$. If the paragraph and examples above look familiar, it is because this is exactly like finding the area under $y= 2 + 0.02x$ between $x=1$ and $x=4$. Only, in this case, the function is $v$ instead of $f$, and the variable $t$ instead of $x$, but it's the same calculation.
It follows that distance is the area between the velocity curve
and the $t$axis. DO: What does it mean
if the velocity is negative? How would that affect the
distance traveled? Sketch a graph of a velocity curve that
is positive for some times and negative for others to help you
see what is happening.
