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CHAPTER 4

CHAPTER 4. THE DEFINITE INTEGRAL. 4.1 Introduction to Area. Finding area of polygonal regions can be accomplished using area formulas for rectangles and triangles. Finding area bounded by a curve is more challenging.

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CHAPTER 4

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  1. CHAPTER 4 THE DEFINITE INTEGRAL

  2. 4.1 Introduction to Area • Finding area of polygonal regions can be accomplished using area formulas for rectangles and triangles. • Finding area bounded by a curve is more challenging. • Consider that the area inside a circle is the same as the area of an inscribed n-gon where n is infinitely large.

  3. Adding infinitely many terms together • Summation notation simplifies representation • Area under any curve can be found by summing infinitely many rectangles fitting under the curve.

  4. 4.2 The Definite Integral • Riemann sum is the sum of the product of all function values at an arbitrary point in an interval times the length of the interval. • Intervals may be of different lengths, the point of evaluation could be any point in the interval. • To find an area, we must find the sum of infinitely many rectangles, each getting infinitely small.

  5. Definition: Definite Integral • Let f be a function that is defined on the closed interval [a,b]. • If exists, we say f is integrable on [a,b]. Moreover, called the definite integral (or Riemann integral) off from a to be, is then given as that limit.

  6. Area under a curve • The definite integral from a to b of f(x) gives the signed area of the region trapped between the curve, f(x), and the x-axis on that interval. • The lower limit of integration is a and the upper limit of integration is b. • If f is bounded on [a,b] and continuous except at a finite number of points, then f is integrable on [a,b]. In particular, if f is continuous on the whole interval [a,b], it is integrable on [a,b].

  7. Functions that are always integrable • Polynomial functions • Sin & cosine functions • Rational functions, provided that [a,b] contains no points where the denominator is 0.

  8. 4.3 First Fundamental Theorem • Let f be continous on the closed interval [a,b] and let x be a (variable) point in (a,b). Then

  9. What does this mean? • The rate at which the area under the curve of function, f(t), is changing at a point is equal to the value of the function at that point.

  10. 4.4 The 2nd Fundamental Theorem of Calculus and the Method of Substitution • Let f be continuous (integrable) on [a,b], and let F be any antiderivative of f on [a,b]. Then the definite integral is

  11. Evaluate

  12. Substitution Rule for Indefinite Integrals • Let g be a differentiable function and suppose that F is an antiderivative of f. Then

  13. What does this remind you of? • It is the chain rule! (from differentiation) • In this case, you have an integral with a function and it’s derivative both present in the integrand. • This is often referred to as “u-substitution” • Let u=function and du=that function’s derivative

  14. Evaluate

  15. Substitution Rule for Definite Integrals • Let g have a continuous derivative on [a,b], and let f be continuous on the range of g. Then where u=g(x):

  16. What does this mean? • For a definite integral, when a substitution for u is made, the upper and lower limits of integration must change. They were stated in terms of x, they must be changed to be the corresponding values, in terms of u. • When this change in the upper & lower limits is made, there is no need to change the function back to be in terms of x. It is evaluated in terms of the upper & lower limits in terms of u.

  17. Evaluate:

  18. 4.5 The Mean Value Theorem for Integrals and the Use of Symmetry • Average Value of a Function: If f is integrable on the interval [a,b], then the average value of f on [a,b] is:

  19. What does this mean? • If you consider the definite integral from over [a,b] to be the area between the curve f(x) and the x-axis, f-average is the height of the rectangle that would be formed over that same interval containing precisely the same area.

  20. Mean Value Theorem for Integrals • If f is continuous on [a,b], then there is a number c between a and b such that

  21. Symmetry Theorem • If f is an even function then • If f is an odd function, then

  22. 4.6 Numerical Integration • If f is continuous on a closed interval [a,b], then the definite integral must exist. However, it is not always easy or possible to find the definite integral. • In these cases, we use other methods to closely approximate the definite integral.

  23. Methods for approximating a definite integral • Left (or right or midpoint) Riemann sums (estimate the area with rectangles) • Trapezoidal Rule (estimate with several trapezoids) • Simpson’s Rule (estimate the area with the region contained under several parabolas)

  24. Summary of numerical techniques • Approximating the definite integral of f(x) over the interval from a to b.

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