5: Integrals
- Page ID
- 208302
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 5 transitions from derivatives to integration, defining the antiderivative as the mathematical undoing of differentiation to recover original functions like position from velocity. By using Riemann sums and Sigma notation, the chapter builds a geometric bridge to calculate the exact area under a curve as a limit of rectangular approximations. This culminates in the Fundamental Theorem of Calculus, which links these areas directly to antiderivatives, providing a universal method for calculating total accumulation and net change.
- 5.1: Antiderivatives
- An antiderivative is the "reverse" of a derivative; if \(F'(x) = f(x)\), then \(F(x)\) is an antiderivative of \(f(x)\). Because the derivative of any constant is zero, we must always add a constant of integration (\(+C\)) to represent the entire family of possible functions.
- 5.2: Areas and Distances
- This section introduces Sigma notation as a compact mathematical shorthand for expressing and calculating long sums of terms using an index. By applying this notation to Riemann sums, you can approximate the area under a curve by summing the areas of \(n\) rectangles with heights determined by specific sample points. As the number of rectangles increases toward infinity, these approximations converge to the exact area, bridging the gap between discrete sums and continuous calculus.
- 5.3: The Definite Integral
- If \(f(x)\) is a function defined on an interval \([a,b]\), the definite integral of f from a to b is given by the limit of approximations of the area under the curve (provided the limit exists). If this limit exists, the function \(f(x)\) is said to be integrable on \([a,b]\), or is an integrable function. The numbers a and b are called the limits of integration; specifically, a is the lower limit and b is the upper limit. The function \(f(x)\) is the integrand, and x is the variable of integra
- 5.4: The Fundamental Theorem of Calculus
- The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this is not always easy.
- 5.5: Indefinite Integrals and the Net Change Theorem
- This section introduces the indefinite integral as the general set of all antiderivatives for a function. The Net Change Theorem then applies this concept to physical rates, stating that the integral of a rate of change (like velocity) over an interval gives the total change in the original quantity (like displacement). This connection allows you to calculate the total accumulation of a value simply by finding the difference between the antiderivative's values at two points.
- 5.6: Substitution
- This section introduces substitution as a technique to reverse the Chain Rule by simplifying a complex integral into a more basic form. By replacing a composite part of the function with a new variable \(u\) and its derivative \(du\), you can transform the integral into one of the standard shortcut formulas. This method is essential for solving integrals where one part of the integrand is the derivative of another.