Capstone Porject Part 3

# Capstone Porject Part 3 - Rima Teymourlouei Math-181-401...

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Rima Teymourlouei Math-181-401 Vaughn Capstone Project Part III: Function 2 y = ax 2 + bx + c Dependent variables: x Independent variables: y Constants: a = 1, b = 3, c = 2 f(x) = x 2 + 3x + 2

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Left, Right, Midpoint Riemann Sums Left Riemann Sum (n = 4): 5 4 ( −16+5𝑘−5 4 ) 2 + 3( −16+5𝑘−5 4 4 𝑘=1 ) + 2 = 10.4688 Right Riemann Sum (n = 4): 5 4 ( −16+5𝑘 4 ) 2 + 3( −16+5𝑘 4 4 𝑘=1 ) + 2 = 10.4688
Midpoint Riemann Sum (n = 4): 5 4 ( −32+10𝑘−5 4 ) 2 + 3( −32+10𝑘−5 4 4 𝑘=1 ) + 2 = 8.51563 Left Riemann Sum (n = 8): 5 8 ( −32+5𝑘−5 8 ) 2 + 3( −32+5𝑘−5 8 8 𝑘=1 ) + 2 = 9.49219

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Right Riemann Sum (n = 8): 5 8 ( −32+5𝑘 8 ) 2 + 3( −32+5𝑘 8 8 𝑘=1 ) + 2 = 9.49219 Midpoint Riemann Sum (n = 8): 5 8 ( −64+10𝑘−5 8 ) 2 + 3( −64+10𝑘−5 8 8 𝑘=1 ) + 2 = 9.00391 The Riemann Sums all provide an approximation of the area under the curve. When the slopes of the function are negative, in this case from the interval of -4 to -1.5, the Left Riemann Sum is an over estimation, the Right Riemann Sum is an under estimation, and the Midpoint Riemann Sum is an under estimation. When the slopes of the function are positive, in this case from the interval
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