SECTION 4-2 (A) Application of the Integral

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) Each rectangle has a height f(x) and a width dx Add the area of the rectangles to approximate the area under the curve

Consider the equation:

3) find the area under the curve from x = 1 to x = 5 using two rectangles of equal width. 153 How can we get a better approximation?

4) For the previous problem use four rectangles 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. Even More Rectangles 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: Width of each rectangle along the x-axis Value of function at a the leftmost endpoint Values of function at intermediate x-values Value of function at second to last endpoint. Excludes the rightmost endpoint

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 2 nd 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

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

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

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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