Riemann Sums Approximate area using rectangles Section 4-2 (a) Riemann Sums Approximate area using rectangles

1) The graph on the right, is of the equation How would you find the area of the shaded region?

2) The graph on the right, is of the equation of a semicircle How would you find the area of the shaded region?

Area of Common known Geometric shapes Triangle – Rectangle – Semicircle – Trapezoid – *Give exact area under the curve

What if the curve doesn’t form a geometric shape? Determine area is by finding the sum of rectangles Use rectangles to approximate the area between the curve and the x – axis: Archimedes (212 BC) Add the area of the rectangles to approximate the area under the curve Each rectangle has a height f(x) and a width dx

Consider the equation:

How can we get a better approximation? 3) find the area under the curve from x = 1 to x = 5 using two rectangles of equal width. 1 5 3 How can we get a better approximation?

4) For the previous problem use four rectangles More rectangles 4) For the previous problem use four rectangles 1 2 3 4 5

Even More Rectangles Suppose we increase the number of rectangles, then the area underestimated by the rectangles decreases and we have a better approximation of the actual area. How can we get an even better approximation?

Rectangles formed by the left-endpoints: Let n be the number of rectangles used on the interval [a,b]. Then the area approximated using the left most endpoint is given by: Value of function at a the leftmost endpoint Value of function at second to last endpoint . Excludes the rightmost endpoint Values of function at intermediate x-values Width of each rectangle along the x-axis

Left end-point rectangles The sum of the areas of the rectangles shown above is called a left-hand Riemann sum because the left-hand corner of each rectangle is on the curve.

Rectangles formed by the right-endpoints: Let n be the number of rectangles used on the interval [a,b]. Then the area approximated using the right most endpoint is given by: Value of function at 2nd x-value. Excludes the leftmost endpoint Value of function at the last endpoint b. Width of each rectangle along the x-axis Values of function at intermediate x-values

Right end-point rectangles The sum of the areas of the rectangles shown above is called a right-hand Riemann sum because the right-hand corner of each rectangle is on the curve.

Circumscribed vs. Inscribed Circumscribed Rectangles: Extend over the curve and over estimate the area Inscribed Rectangle: Remain below the curve and under estimate the area http://webspace.ship.edu/msrenault/GeoGebraCalculus/integration_riemann_sum.html

Left-Endpoint Approximations Circumscribed: when the function is decreasing Inscribed: when the function is increasing Right-Endpoint Approximations is increasing Inscribed: when the function is decreasing

Upper and Lower Sums Lower Sum Upper Sum: The sum of the The sum of the Inscribed rectangles circumscribed rectangles

5) Approximate the area under the curve on the interval [ 0 ,4] and n = 4 using a right hand Reimann sum

6) Find the upper and lower sums of on the interval [ 0 ,3] and n = 3

7) Use left endpoints to approximate the area under the curve on the interval [ 0 ,3] and n = 3

8) Use right endpoints to approximate area under on the interval [ 0 ,2] using 8 rectangles

9) Use left endpoints to approximate area under on the interval [ 0 ,2] using 8 rectangles

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