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 LimitsClose is good enoughDefinition Onesided Limits How can a limit fail to exist? Infinite Limits and Vertical Asymptotes Summary Limit Laws and ComputationsA summary of Limit LawsWhy do these laws work? Two limit theorems How to algebraically manipulate a 0/0? Limits with fractions Limits with Absolute Values Limits involving Rationalization 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 Examples of continuous functions Limits at InfinityLimits at infinity and horizontal asymptotesLimits at infinity of rational functions Which functions grow the fastest? Vertical asymptotes (Redux) Toolbox of graphs Rates of ChangeTracking changeAverage and instantaneous velocity Instantaneous rate of change of any function Finding tangent line equations Definition of derivative 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 FunctionsTwo important LimitsSine and Cosine 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 DifferentiationIntroduction and ExamplesDerivatives 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 ReviewBasic Building BlocksAdvanced 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 Fermat's Theorem Howto The Mean Value and other TheoremsRolle's TheoremsThe Mean Value Theorem Finding $c$ $f$ vs. $f'$Increasing/Decreasing Test and Critical NumbersHowto The First Derivative Test Concavity, Points of Inflection, and the Second Derivative Test Indeterminate Forms and L'Hospital's RuleWhat does $\frac{0}{0}$ equal?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 AntiderivativesAntiderivatives and PhysicsSome formulas Antiderivatives are not Integrals The Area under a curveThe Area Problem and ExamplesRiemann Sums Notation Summary Definite IntegralsDefinitionProperties What is integration good for? More Examples The Fundamental Theorem of CalculusThree Different QuantitiesThe Whole as Sum of Partial Changes The Indefinite Integral as Antiderivative The FTC and the Chain Rule 
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: $$ \hbox{Rate} \times \hbox{Time} = \hbox{Distance}.$$ 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.
The answer in Example 2 is 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 $(41)v(2.5)=3(2.05)=6.15$. 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 $t=4$. By now you've probably noticed that 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. In general,
