Chapter 5 Integral

Estimating with Finite Sums

Approach

Approach (2)

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

Therefore

Midpoint approach

Conclusions:

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

Average Value of a Nonnegative Function

Sigma Notation and Limits of Finite Sums

Limits of Finite Sums Solution: We start by subdividing [0, 1] into n equal width subintervals The lower sum of rectangular is :

Riemann Sums

Riemann Sums(2) the width of the kth subinterval is

Riemann Sums(3)

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.

The Definite Integral

Notation and existence of definite Integrals

Properties of Definite Integrals

Average Value of a Continuous Function Revisited

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (2) But remember this

Indefinite Integrals and the Substitution Rule Symbol 

Substitution: Running the Chain Rule Backwards

Definite Integrals of Symmetric Functions

Areas Between Curves

But, be careful with this circumstances

Integration with Respect to y

Example : previous problem, but integration respects to y

Combining Integrals with Formulas from Geometry