ProjectI(TI 84) - Math1242 Project I(TI 84 Name Riemann...

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Math1242 Project I (TI 84) Name: Riemann Sums and Definite Integrals The area under the graph of a positive function is given by the definite integral of the function. The definite integral can be approximated by the following sums: Left Riemann Sum : )] ( .......... ) ( ) ( [ ) ( 1 1 0 n n b a x f x f x f x L dx x f Right Riemann Sum : )] ( .......... ) ( ) ( [ ) ( 2 1 n n b a x f x f x f x R dx x f Midpoint Rule : )] ( .......... ) ( ) ( [ ) ( 1 2 1 n n b a x f x f x f x M dx x f Trapezoidal Rule : )] ( ) ( 2 .......... ) ( 2 ) ( [ 2 ) ( 1 1 0 n n n b a x f x f x f x f x T dx x f Simpson’s Rule : )] ( ) ( 4 .......... ) ( 2 ) ( 4 ) ( [ 3 ) ( 1 2 1 0 n n n b a x f x f x f x f x f x S dx x f Where n a b x , x i a x i , i x the midpoint of the i th subinterval , and n is even for Simpson’s Rule. It turns out that the Trapezoidal approximation 2 n n n R L T and Simpson’s approximation 3 2 2 / 2 / n n n M T S In all of these methods we get more accurate approximations when we increase the value of n . The Error in using an approximation is: Error = Actual value of the integral - Approximation = b a dx x f ) ( - Approximation Error Bounds for Midpoint and Trapezoidal Rules: Suppose that 1 ) ( K x f for b x a . Then | E M | 2 3 1 24 ) ( n a b K and | E T | 2 3 1 12 ) ( n a b K Error Bounds for Simpson’s Rules: Suppose that 2 ) 4 ( ) ( K x f for b x a . Then | E S | 4 5 2 180 ) ( n a b K These Error Bounds are very useful to estimate the errors and the accuracy of the approximations without having to find the value of these approximations, especially for large n ’s. These Error Bounds are also helpful in estimating the number of partitions required to guarantee a specific accuracy when approximating an integral.
The above Approximating Sums can be found using a simple program on the calculator called RIEMANN . We can use the RIEMANN program to approximate the following integral: dx x 2 0 2 3 Press Y = and set Y 1 = 3X 2 . Press 2 nd QUIT to go to the home screen. Press PRGM to display the PRGM EXEC menu. Select the program’s name and press ENTER to execute the program.

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