Calculus Fundamentals: Concepts, Theorems, and Examples

1. Example • We can also evaluate a definite integral by interpretation of definite integral. • Ex. Findbyinterpretation 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,

2. Example • Ex. Byinterpretation of definite integral, find • Sol. (1) (2)

3. 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].

4. 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,

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

6. 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):

7. 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

8. 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].

9. 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

10. Function defined by definite integrals with varying limit • Suppose f is integrable on [a,b]. For any given the definite integralis 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

11. 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].

12. 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,

13. 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

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

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

16. Example • Ex. Find derivative • Sol. • Ex. Find if • Sol.

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

18. Example • Ex. Find the limit • Sol.

19. 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

20. 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.

21. 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

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

23. Example • Ex. Evaluate • Sol. • Ex. Evaluate • Sol.

24. Example • Anything wrong in the following calculation?

25. 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.

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