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# Class Note for MATH 1314 with Professor Flagg at UH 2

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Date Created: 02/06/15
Page 1 of6 Math 1314 Lesson 18 Area and the De nite Integral We are now ready to tackle the second basic question of calculus 7 the area question We can easily compute the area under the graph of a function so long as the shape of the region conforms to something for which we have a formula for geometry Example 1 Suppose f x 5 Find the area under the graph of f x from x 0 to x 4 Approximating Area Under a Curve Now suppose the area under the curve is not something whose area can be easily computed We ll need to develop a method for finding such an area Example 2 Here we ll draw some rectangles to approximate the area under the curve We can find the area of each rectangle then add up the areas to approximate the area under the curve Math 1314 Lesson 18 Page 1 of6 Page 2 of 6 Example 3 Next We ll increase the number of rectangles Example 4 And We ll increase the number of rectangles again 1W Math 1314 Lesson 18 Page 2 of6 Page 3 of6 What you should see is that as the number of rectangles increases the area we compute using this method becomes more accurate The Area Under the Graph of a Function Let f be a nonnegative continuous function on 61 b Then the area of the region under the graph of f is given by A1iIgfxlfxZfxnAx b a n where x1 x2 x are arbitrary points in the interval a b of equal width Ax The sums of areas of rectangles are called Riemann sums and are named after a German mathematician Example 4 Use left endpoints and 4 subdivisions of the interval to approximate the area under f x 2x2 l on the interval 0 2 Math 1314 Lesson 18 Page 3 of6 Page 4 of6 Example 5 Use right endpoints and 4 subdivisions of the interval to approximate the area under f x 2x2 1 on the interval0 2 Example 6 Use midpoints and 4 subdivisions of the interval to approximate the area under f x 2x2 1 on the interval 0 2 Math 1314 Lesson 18 Page 4 of6 Page 5 of6 Example 7 Suppose f x 1 3x Approximate the area under the graph of f on the interval 0 12 using 6 subdivisions and left endpoints The De nite Integral Letfbe de ned on a b If limfx1 fx2 fxn Axexists for all choices of b a representative points in the n subintervals of a b of equal width Ax then this n limit is called the de nite integral of f from a to b The de nite integral is noted by r7 fxdxlimfx1 fx2 fxn Ax The number a is called the lower limit of integration and the number b is called the upper limit of integration A function is said to be integrable on a b if it is continuous on the interval a b Math 1314 Lesson 18 Page 5 of6 Page 6 of6 The de nite integral ofa nonnegative function The de nite integral of a general function From this section you should be able to Explain the procedure used to approximate area under a curve Use Riemann sums to approximate the area under a curve using right endpoints left endpoints or midpoints Explain what we mean by de nite integral of a nonnegative function or a general function Math 1314 Lesson 18 Page 6 of6

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