Understanding the Definite Integral in Calculus
Learn about the concept of the definite integral in calculus, where we define and calculate the signed area between a function and an interval. Explore the symbols, theorem, examples, and solutions to grasp the fundamentals of integration.
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Presentation Transcript
In this sequence we define the definite integral of a function ? on an interval ?. Let us assume, for the time being, that ?(?) is defined and continuous on the closed, finite interval [?,?]. We no longer assume that the values of ? are nonnegative. The symbol for the definite integral of ?(?) over the interval [?,?] is ?? ? ?? ? which is read as the integral from ? to ? of ? of ? with respect to ?.
The component parts in the integral symbol also have names as shown in the figure below.
The following theorem says that if a function is continuous on a finite closed interval, then it is integrable on that interval, and its definite integral is the net signed area between the graph of the function and the interval. Theorem: If a function ? is continuous on an interval [?,?], then ? is integrable on [?,?], and the net signed area ? between the graph of ? and the interval ?,? is ?? ? ??. ? = ?
Example: Sketch the region whose area is represented by the definite integral ???? ? and evaluate the integral using an appropriate formula from geometry. Solution: The graph of the integrand is the horizontal line ? = ?, so the region is a rectangle of height ? extending over the interval from to . Thus, ???? area of rectangle = ? ? = ? ?
Example: Sketch the region whose area is represented by the definite integral ? ? ???? ? and evaluate the integral using an appropriate formula from geometry. Solution: The graph of ? ??is the upper semicircle of radius ?, centered at the origin, so the region is the right quarter-circle extending from ? = ? to ? = ?.
? ? ???? area of quarter-circle =? Thus, ? ? (?)?? =? ? Example: Evaluate ?? ? ?? 1. ? ?? ? ?? 2. ?
Solution: The graph of ? = ? ? is shown in the figure, and we leave it for you to verify that the shaded triangular regions both have area ?/?. Over the interval [?,?] the net signed area is ?? ??=? and over the interval [?,?] the net signed area is ??= ? ? Thus, ?? ? ?? = ? and ? ? ? ?= ? ?? ? ?? = ? ?. ?
Example: Use the areas shown in the figure to find: Solution: 1. Over the interval [?,?], the net signed area is 0.8 and hence ? ?? ? ?? = 0.8.
Over the interval [?,?] , the net signed area is ?.? and hence ? 2. ?? ? ?? = ?.?. Over the interval [?,?] , the net signed area is ?.? ?.? = ?.? and hence ? 3. ?? ? ?? = ?.?. Over the interval [?,?] , the net signed area is ?.? ?.? + ?.? = ?.? and hence ? 4. ?? ? ?? = ?.?.
Properties of the Definite Integral: ?? ? ?? = ?. 1. If ? is in the domain of ?, we define ? 2. If ? is integrable on [?,?], then we define ? = ? ?? ? ?? ?? ? ?? 3. If ? and ? are integrable on [?,?] and if ? is a constant, then ??, ? + ?, and ? ? are integrable on [?,?] and
4. If ? is integrable on a closed interval containing the three points a,? , and ? (no matter how the points are ordered), then ?? ? ?? = ? ?? ? ?? + ? ?? ? ?? ? 5. If ? is integrable on [?,?] and ?(?) 0 for all in [?,?] , then ?? ? ?? 0 ? and if ?(?) 0 for all ? in [?,?], then ?? ? ?? 0 ?
6. If ? and ? are integrable on [?,?] and ?(?) ?(?) for all ? in [?,?], then ?? ? ?? ? ?? ? ?? ? ? Example: Find ? ?? ? ?? = ? and ? Solution: ? ? ? + ?? ? ? ? + ?? ? if ?? ? ?? = ?. ? ? ? = ? ? ?? + ? ? ? ?? ? ? ? = ? + ? ? = ?
?? ? ?? if Example: Find ? ?? ? ?? = ? and ? Solution: ? ? ? ?? = ?? ? ?? = ?. ? ? ? ? ? ?? + ? ? ?? ? ? ? ? ? = ? + ? ? ?? ? ? ? ? ?? = ? ?
Example: Determine whether the value of the integral ? ? ? ??? ? is positive or negative. Solution: Since ? > ? and ? ? < ? on [?,?] then ? ? ?< ? ??? ? ? ? ??? < ? ?
