ch05 - 90 Chapter 5: Integration Summary: The central...

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Unformatted text preview: 90 Chapter 5: Integration Summary: The central concept introduced in this chapter is that of integration and how to find the area under a curve. There are two varieties of integration represented by indefinite integrals and definite integrals. One of the main features of this chapter, the Fundamental Theorem of Calculus, shows how integrals are tied very closely to the notion of a derivative of a function. In fact, the indefinite integral is actually a way of representing the antiderivative of a function and so integration is often seen as the opposite of differentiation. Towards the end of the chapter, definite integrals are related to rectilinear motion and the concepts of position, velocity and speed. At the end of the chapter, log- arithmic functions are redefined using definite integrals. This gives another per- spective from which to view logarithmic functions and their inverses, exponential functions. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Approximate a definite integral using the rectangle method (§5 . 1). 2. Find general antiderivatives of functions by evaluating indefinite integrals (§5 . 1). 3. Evaluating basic indefinite integrals (§5 . 2). 4. Evaluate constants of integration given an initial condition (§5 . 2). 5. Use u-substitution to evaluate indefinite integrals (§5 . 3). 6. Represent the area under a curve as a limit using sigma notation (§5 . 4). 7. Numerically approximate the area under a curve (§5 . 4). 8. Identify definite integrals as representing the net signed area between a function and an interval (§5 . 4 , 5 . 5). 9. Identify definite integrals as limits of Riemann sums (§5 . 5). 10. Evaluate definite integrals geometrically (§5 . 5 , 5 . 6). 91 92 11. Understand the difference between net-signed area (§5 . 4 , 5 . 5) and total area (§5 . 6). 12. Evaluate definite integrals using the Fundamental Theorem of Calculus (§5 . 6). 13. Define functions using integrals and be able to take their derivatives by the Fundamental Theorem of Calculus (§5 . 6 , 5 . 10). 14. Calculate displacement of a particle by a definite integral of the velocity (§5 . 7). 15. Find the average value of a function (§5 . 8). 16. Evaluate definite integrals that require u-substitution (§5 . 9). 17. Define the logarithmic function using an integral (§5 . 10). 5.1 An Overview of the Area Problem PURPOSE: To find the area under a curve over an interval. The primary problem considered in this section is to find the area under a curve over a particular interval. Specifically, only positive functions are considered here (or at least intervals where the functions considered are positive). Two pri- mary methods of calculating areas are introduced: using rectangles and using antiderivatives....
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ch05 - 90 Chapter 5: Integration Summary: The central...

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