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 
The Area Problem and ExamplesIntegration is a process for computing bulk quantities by estimating
and adding up smaller pieces. The mantra (and the 5th Pillar of Calculus) is:
The idea works in many different settings. In this learning module we will develop the simplest application of integration, namely finding the area under a curve.
The answer in Example 2 is a little bit too small, since the curve goes up from $y=2.02$ at $x=1$ to $y=2.08$ at $x=4$. If we wanted an overestimate of the area, we would use the value of $f(x)$ at the right endpoint instead: $(41)f(4)=3(2.08)=6.24$. And if we wanted a more accurate guess, we might use the midpoint $f(2.5)=2.05$ and estimate the area as $(41)f(2.5)=3(2.05)=6.15$. But whether you use the left endpoint, the right endpoint, or the midpoint, you get roughly the same answer, since the function just isn't changing very much between $x=1$ and $x=4$. So how do you find the area under a curve that is changing a lot? For instance, how would you find the area under $y=x^2$ between $x=1$ and $x=1$? The answer is to break the interval $[1,1]$ up into pieces, each so small that the function doesn't change much in that piece. Then add up the pieces to get a good estimate of the total. As with most of calculus, the exact answer is then the limit of these approximate answers. The animation below shows how this area can be approximated by adding up narrower and narrower rectangles, with $x_i^*$ being the left/mid/right endpoint. Main Ideas and Examples
