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 [x i-1,x i ] (i=1, ,n) with x 0 =a and x n =b; then, approximate each subregion S i by a rectangle since f(x) does not change much and can be treated as a constant in each subinterval [x i-1,x i ], that is, S i ¼ (x i -x i-1 )f(  i ), where  i is any point in [x i-1,x i ]; 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 {x i } are arbitrarily chosen, the second is that the sample points {  i } are arbitrarily taken too. means, no matter how {x i } and {  i } 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 d i in each subinterval [t i-1,t i ] by d i ¼ (t i -t i-1 )v(  i ) 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 from a to b, which is denoted by

6. Remark The usual way of partition is 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 of integration; 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