How to understand the MEAN VALUE THEOREM for INTEGRALS – CALCULUS 2
How to understand the MEAN VALUE THEOREM for INTEGRALS – CALCULUS 2

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Example We can also evaluate a definite integral by interpretation of definite integral. Ex. Find by interpretation of definite integral. Sol. By the interpretation of definite integral, we know the definite integral is the area of the region under the curve from 0 to a. From the graph, we see the region is a quarter disk with radius a and centered origin. Therefore,

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Example Ex. By interpretation of definite integral, find Sol. (1) (2)

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Properties of definite integral Theorem(linearity of integral) Suppose f and g are integrable on [a,b] and are constants, then is integrable on [a,b] and Theorem(product integrability) Suppose f and g are integrable on [a,b], then is integrable on [a,b].

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Properties of definite integral Theorem(additivity with respect to intervals) Remark In the above property, c can be any number, not necessarily between a and b. When the upper limit is less than the lower limit in the definite integral, it is understood as Especially,

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Comparison properties of integral 1. If for then 2. If for then 3. If for then 4.

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Estimation of definite integral Ex. Use the comparison properties to estimate the definite integral Sol. Denote Then when Letting we get the only critical number By the closed interval method, we find the range for f(x):

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Mean value theorems for integrals Second mean value theorem for integrals Let g is integrable and on [a,b]. Then there exists a number such that Proof. Let Since we have and Hence or By intermediate value theorem

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Mean value theorems for integrals First mean value theorem for integrals Let then there exists a number such that Remark. We call the mean value of f on [a,b].

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Example Ex. Suppose and Prove that such that Proof. By the first mean value theorem for integrals, there exists such that Thus By Rolle’s theorem, such that

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Function defined by definite integrals with varying limit Suppose f is integrable on [a,b]. For any given the definite integral is a number. Letting x vary between a and b, the definite integral defines a function: Ex. Find a formula for the definite integral with varying limit Sol. By interpretation of definite integral, we have

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Properties of definite integral with varying limit Theorem(continuity) If f is integrable on [a,b], then the definite integral with varying limit is continuous on [a,b].

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The fundamental theorem of calculus (I) The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a,b], then the definite integral with varying limit is differentiable on [a,b] and Proof is between x and as and Therefore,

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Definite integral with varying limits The definite integral with varying lower limit is Since we have The most general form for a definite integral with varying limits is To investigate its properties, we can write it into the sum of two definite integrals with varying upper limit

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Definite integral with varying limits By the chain rule, we have the formula

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Example Ex. Find derivatives of the following functions Sol. (2) Let by chain rule,

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Example Ex. Find derivative Sol. Ex. Find if Sol.

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Example Ex. Find the limit Sol. By L’Hospital’s Rule and equivalent substitution, Question:

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Example Ex. Find the limit Sol.

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Example Ex. Suppose b>0, f continuous and increasing on [0,b]. Prove the inequality Sol. Let Then F(0)=0 and when This implies F(t) is increasing, thus

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Example Ex. Suppose f is continuous and positive on [a,b]. Let Prove that there is a unique solution in (a,b) to F(x)=0. Sol.

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Fundamental theorem of calculus (II) The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a,b] and F is any antiderivative of f, then Proof Let then g is an antiderivative of f. So F(x)=g(x)+C. Therefore, Remark The formula is called Newton-Leibnitz formula and often written in the form

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Example Ex. Evaluate Sol. Ex. Find the area under the parabola from 0 to 1. Sol.

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Example Ex. Evaluate Sol. Ex. Evaluate Sol.

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Example Anything wrong in the following calculation?

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Differentiation and integration are inverse The fundamental theorem of calculus is summarized into The first formula says, when differentiation sign meets integral sign, they cancel out. The second formula says, first differentiate F, and then integrate the result, we arrive back to F.

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Homework 12 Section 5.1: 21 Section 5.2: 22, 37, 53, 59, 67 Section 5.3: 18, 50, 54, 62

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