Friday+February+4+Accumulation+Functions+Day+2

(Charles) How is a midpoint approximation different from a left or right hand approximation?

Midpoint approximations are made by creating rectangles with the height as tall as the function at the point in the middle of each intervals. (the intervals are made by delta x) Midpoint approximation is more accurate than left or right because some of the rectangle created goes over the curve while some is under. Left and right approximations rectangles either go completely under the curve, or over. Sometimes. We'll look at some situations where this is not true! The two parts in the midpoint approximation rectangle almost cancel each other out. Midpoint approximations are still over approximations or under approximation but it depends on concavity. Nice job!!

(Joe) What does the definite integral represent, geometrically?

The integral represents the area that is under the function. We use the area of the function to approximate the change in output for the derivative of the first function. The general equation that we use to determine the area of the function is a Riemann sum, or the equation for definite integrals which is: The equation of the definite integral makes sense as an estimate of the area of the function because f(x) represents the height and dx represents the width of the rectangles that we use to approximate the area. However, if the function that we are approximating is curved, a rectangle would not perfectly fit underneath. This is why there are left and right hand approximations. Left hand uses the left side of the triangle as the height, while the right hand uses the right side. We attempt to reach a better estimate of the area underneath the curve by making dx as close to 0 as possible. wonderful explanation!!

(Han) What does the definite integral represent, in terms of functions? The definite integral represents the limit of the sum (as //n// subintervals reaches infinity) of the areas of approximating rectangles within the graph of the function, with the height of the //i//th rectangle to be the value of f at any number x//i//* in the ith subinterval [x//i//-1, x//i//]. The sum of the areas of these approximating rectangles can be written as the sigma notation of f of f(x//i//*)dx (starting from //i//=1 to //n//), or can be written as the sigma notation of f(x1)dx+f(x2)dx+f(x3)dx...+f(xn)dx (once again starting from i=1 to n). So if f is a function with the intervals of [a,b], which is divided into //n// subintervals of equal width dx=(b-a)///n//, and if x0(=a), x1, x2,...,x//n//(=b) are the endpoints of these subintervals with x1*, x2*,..,x//n//* being any sample points in these subintervals (so that x//i//* lies in the //i//th subinterval [x//i//-1, x//i//]), then the definite integral of f from a to b is the limit as n approaches infinity of f(x//i//*)dx provided that this limit exists. The sum f(x//i//*)dx (with //i//=1 as the starting point and //n// being the endpoint) that occurs in the definite integral is called a Riemann sum. So the definite integral of an integrable function can be approximated to within any desired degree of accuracy by a Riemann sum. whew!! very good!!

Homework page 376 numbers 7, 9, 11, 33, 34, 35, 37, 39, 48, 50