Both approach are called Upper sum because they are obtained by taking the height of each rectangle as the maximum (uppermost) value of ƒ(x) for x a point in the base interval of the rectangle. Now, we will be using what so called lower sum
Distance travelled Suppose we know the velocity function y(t) of a car moving down a highway, without changing direction, and want to know how far it traveled between times t=a and t=b If we already known an antiderivative F(t) of v(t) we can find the car’s position function s(t) by setting s(t)=F(t)+C. The travelled distance is s(b)-s(a) How to calculate in case we have no formula s(t)? We need an approach in calculating s(t)
approach Subdivide the interval [a, b] into short time intervals on each of which the velocity is considered to be fairly constant. distance = velocity x time Total distance
Riemann Sums (4) Among three figures, which one gives us the most accurate calculation?
Riemann Sums (5) In previous calculation, we can improve accuracy by increasing number of interval (n). However, in Reimann sum, we can go to more accurate calculation by making |P| goes to zero We define the norm of a partition P, written |P| to be the largest of all the subinterval widths. If |P| is a small number, then all of the subintervals in the partition P have a small width.