Introduction to integrals

1. Introduction to integrals • Integral, like limit and derivative, is another important concept in calculus • Integral is the inverse of differentiation in some sense • There is a connection between integral calculus and differentiation calculus. • The area and distance problems are two typical applications to introduce the definite integrals

2. The area problem • Problem: find the area of the region S with curved sides, which is bounded by x-axis, x=a, x=b and the curve y=f(x). • Idea: first, divide the region S into n subregions by partitioning [a,b] into n subintervals [xi-1,xi] (i=1,L,n) with x0=a and xn=b; then, approximate each subregion Si by a rectangle since f(x) does not change much and can be treated as a constant in each subinterval [xi-1,xi], that is, Si¼(xi-xi-1)f(xi), where xi is any point in [xi-1,xi]; last, make sum and take limit if the limit exists, then the region has area

3. Remark • In the above limit expression, there are two places of significant randomness compared to the normal limit expression: the first is that the nodal points {xi} are arbitrarily chosen, the second is that the sample points {xi} are arbitrarily taken too. • means, no matter how {xi} and {xi} are chosen, the limit always exists and has same value.

4. The distance problem • Problem: find the distance traveled by an object during the time period [a,b], given the velocity function v=v(t). • Idea: first, divide the time interval [a,b] into n subintervals; then,approximate the distance di in each subinterval [ti-1,ti] by di¼(ti-ti-1)v(xi) since v(t) does not vary too much and can be treated as a constant; last, make sum and take limit if the limit exists, then the distance in the time interval [a,b] is

5. Definition of definite integral • We call a partition of the interval [a,b]. is called the size of the partition, where are called sample points. is called Riemann sum. • Definition Suppose f is defined on [a,b]. If there exists a constant I such that for any partition p and any sample points the Riemann sum has limit then we call f is integrable on [a,b] and I is the definite integral of f froma to b, which is denoted by

6. Remark • The usual way of partitionis the equally-spaced partition so the size of partition is In this case is equivalent to • Furthermore, the sample points are usually chosen by or thus the Riemann sum is given by

7. Example • Ex. Determine a region whose area is equal to the given limit (1) (2)

8. Definition of definite integral • In the notation a and b are called the limits ofintegration; a is the lower limit and b is the upper limit; f(x) is called the integrand. • The definite integral is a number; it does not depend on x, that is, we can use any letter in place of x: • Ex. Use the definition of definite integral to prove that is integrable on [a,b], and find

9. Interpretation of definite integral • If the integral is the area under the curve y=f(x) from a to b • If f takes on both positive and negative values, then the integral is the net area, that is, the algebraic sum of areas • The distance traveled by an object with velocity v=v(t), during the time period [a,b], is

10. Example • Ex. Find by definition of definite integral. • Sol. To evaluate the definite integral, we partition [0,1] into n equally spaced subintervals with the nodal points Then take as the sample points. By taking limit to the Riemann sum, we have

11. Example • Ex. Express the limit into a definite integral. • Sol. Since we have with Therefore, The other solution is

12. Example • Ex. If find the limit • Sol.

13. Exercise 1. Express the limits into definite integrals: (1) (2) 2. If find