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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

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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

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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.

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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

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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

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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

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

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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

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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

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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

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

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Example Ex. If find the limit Sol.

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

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