6: Applications of Integration
- Page ID
- 175572
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter, we use definite integrals to calculate the force exerted on the dam when the reservoir is full and we examine how changing water levels affect that force. Hydrostatic force is only one of the many applications of definite integrals we explore in this chapter. From geometric applications such as surface area and volume, to physical applications such as mass and work, to growth and decay models, definite integrals are a powerful tool to help us understand and model the world around us.
- 6.1: Volumes of Revolution - The Disk and Washer Methods
- This section covers methods for determining volumes of solids by slicing, specifically using the Disk and Washer Methods. It explains how to set up integrals based on the cross-sectional areas perpendicular to an axis and includes examples to illustrate these techniques. The section emphasizes understanding the geometric foundations of these methods and applying them to find volumes of various solids, such as those generated by rotating curves around an axis.
- 6.2: Volumes of Revolution - Cylindrical Shells
- This section explains how to find volumes of solids of revolution using the Method of Cylindrical Shells. It covers setting up integrals for volumes by revolving a region around an axis, emphasizing the importance of radius and height in forming cylindrical shells. Examples illustrate the application of the method in different scenarios, providing a visual and algebraic understanding of this approach to volume calculation.
- 6.3: Arc Length of a Curve and Surface Area
- This section covers the calculation of the arc length of a curve and the surface area of solids of revolution using integration. It explains the formulas for arc length in terms of integrals and extends these concepts to find surface areas by revolving a curve around an axis. The section includes detailed examples and applications, illustrating how to set up and evaluate the necessary integrals for these geometric properties.
- 6.4: Work
- This section focuses on calculating the work done by a force over a distance using integration. It explains the concept of work in physics, describes the basic formula W=∫ F(x)dx, and provides examples of varying forces, such as springs and gravitational forces. The section includes applications of integration to solve real-world problems related to work, highlighting how to set up and evaluate integrals in different scenarios.
- 6.5: Hydrostatic Force and Pressure
- This section covers hydrostatic force and pressure, explaining how to calculate the force exerted by a fluid at rest on a surface using integration. It introduces the concepts of fluid pressure, force, and depth, and provides formulas for computing these quantities. The section includes practical examples, such as finding the force on submerged plates or walls, and emphasizes setting up integrals based on geometric and physical principles.
- 6.6: Moments and Centers of Mass
- This section discusses moments and centers of mass, using integration to calculate the balance point of a system of masses. It explains how to find the moments about an axis and the center of mass for planar objects and systems with variable density. The section covers the formulas and applications, providing examples that illustrate the concepts of mass distribution in physical systems.
- 6.7: The Mean Value Theorem for Integrals
- This section introduces the Mean Value Theorem for integrals, which states that for a continuous function over a closed interval, there is at least one point where the function's value equals the average value over the interval. It explains how to apply the theorem and its significance in integration, providing examples to illustrate its practical use in calculating average values and related integrals.
- 6.9: Areas Between Curves
- This section covers how to calculate the area between curves using definite integrals. It explains the process of setting up and evaluating integrals to find the area, focusing on functions defined over a common interval. The section includes examples of horizontal and vertical areas, discussing both simple and more complex cases where the curves intersect. Techniques for determining the limits of integration are also highlighted.
- 6.10: Determining Volumes by Slicing
- This section covers methods for determining volumes of solids by slicing, specifically using the Disk and Washer Methods. It explains how to set up integrals based on the cross-sectional areas perpendicular to an axis and includes examples to illustrate these techniques. The section emphasizes understanding the geometric foundations of these methods and applying them to find volumes of various solids, such as those generated by rotating curves around an axis.
Thumbnail: A region between two functions.