AP Calculus BC Review: Riemann Sum Helena Hu, Athena Liu, Cecilia Bao, Brenda Nguyen, and Candice Chung

What is Riemann Sum Riemann Sums are the approximations of the definite integral found by dividing the area into x strips, and approximating the area of each strip by using geometric figures and adding them all up. There are four methods to approximate the area under the curve. Right Riemann Sum (Rectangle) Left Riemann Sum (Rectangle) Midpoint Riemann Sum (Rectangle) Trapezoidal Riemann Sum (Trapezoid) Riemann is just an method for approximation of the area under the curve.

Underestimate and Overestimate Riemann sum is a method for approximation; therefore, the rectangle area you calculate will be either an overestimation or underestimation. This is a common question type which the answer you can easily observed by looking at a graph.

Left Riemann Sum Visual Representation Time (s) 1 2 3 4 5 Velocity (m/s) 9 16 25 Short Cut formula: A= ΔX (y1+y2+y3+y4...) This is a representation of left Riemann Sum. The approximation of the area under the curve start from the left to the right. A rectangle is drawn according to the y-value of each test point as the height and the difference in x-value as the width. All the difference in x is the same. The area is an underestimation in this graph. I have animation in this slide so please don't mess around with the graph

Right Riemann Sum Visual Representation Time (s) 1 2 3 4 5 Velocity (m/s) 9 16 25 Short cut formula: A= Δx (...y5+y4+y3+y2) This is a Right Riemann Sum. The rectangle is drawn from right to the left. The height of each triangle is the y-value, and the width is the difference in x-value. The approximation is an overestimation in this graph.

Midpoint Riemann Sum Visual Representation Time (s) 1 2 3 4 5 Velocity (m/s) 9 16 25 Time (s) 1.5 2.5 3.5 4.5 5.5 Velocity (m/s) 2.25 6.25 12.25 20.25 30.25 Midpoint Riemann Sum Formula: A=Δx(y1.5+y2.5+y3.5+y4.5…) The midpoint Riemann Sum is relatively more accurate comparing to the left and right Riemann Sum. The difference in x value is constant in this graph. The height of the rectangle is determined by the y-value at the midpoint tested.

Trapezoidal Riemann Sum Visual Representation Time (s) 1 2 3 4 5 Velocity (m/s) 9 16 25 Area for Trapezoid formula: A= (½)(a+b)h A=(½)Δx((y1+y2)+(y2+y3)+(y3+y4)+(y4+y5)) A=(½)Δx(y1+2(y2+y3+y4)+y5) Usually, the Trapezoidal Riemann Sum approximation is more accurate compare to left and right Riemann Sum. Just apply the trapezoid area formula to find the area under the curve. The height can be easily find by the difference between each x-value test point. I have animation in this slide so please don't mess around with the graph

Using Rectangles to Find Area Under the Curve Approximate by using four subintervals of equal

What is the common mistake? Although trapezoidal approximation is more accurate than left and right riemann sums, it is still more or less than the true value. We can infer that the area of a trapezoid is less than the true area if the graph of f is concave down, but is more than the true value if the graph of f is concave up.

Example: try to Solve it!!! The following table is given for x and y values. Use the trapezoidal Riemann Sum with 5 intervals indicated by the data table to approximate definite integral 2 to 12. Is the approximation an overestimation or underestimation? x 2 4 6 8 10 12 f(x) 13 15 14 9 3

Example: The following table is given for x and y values. Use right-hand Riemann sum with 3 sub intervals to approximate the definite integral from 3 to 13. Is the approximation an overestimation or underestimation? x 3 5 7 9 11 13 f(x) 1.73 2.24 2.65 3.32 3.60

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