Chapter 5Integral

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