1. LET’S GET INTEGRAL! By: Diana Luo Naralys Batista Janet Chen

2. Definitions

3. Limit Definition of an Integral

4. Mean Value Theorem

5. Average Value of a Function

6. On a snowy winter day, Janet makes hot chocolate for some friends. As she turns off the fire of the boiling pot of hot chocolate, the hot chocolate instantly cools down. The hot chocolate I modeled by a differentiable function of F for where t is measured in minutes and temperature f(t) measured in Fahrenheit. The values of f(t) at selected values of time t are shown in the table above. a) Use the data in the table to approximate the rate of temperature of the hot chocolate changing at time, t =7. Show the computation that leads to your answer. b) Using the correct units, explain the meaning of dt in the context of this problem. Use trapezoidal sum with the four subintervals indicated by the table to estimate dt. c) Evaluate. Using the correct units, explain the meaning of the expression in the context of this problem. d) At time, t =0, the cookies that Janet also made, were removed from the oven at the temperature of 100 F. The temperature of the cookies at the time t is modeled by the differentiable function C for which is known that. Using the given models, at time t=10, how much cooler are the cookies than the hot chocolate?

7. a) Use the data in the table to approximate the rate of temperature of the hot chocolate changing at time, t =7. Show the computation that leads to your answer.

8. b) Using the correct units, explain the meaning of dt in the context of this problem. Use trapezoidal sum with the four subintervals indicated by the table to estimate dt. dt is the average temperature, in degrees Fahrenheit, of the hot chocolate from. The Trapezoidal Approximation: = [188+264+312+69]= 833

9. c) Evaluate. Using the correct units, explain the meaning of the expression in the context of this problem. This is the change, in degree Fahrenheit, of the temperature of the hot chocolate for minutes. This indicates that the hot chocolate has cooled by in that time interval because of the ” shows that the temperature is decreasing. “-“-

10. d) At time, t =0, the cookies that Janet also made, were removed from the oven at the temperature of 100 F. The temperature of the cookies at the time t is modeled by the differentiable function C for which is known that. Using the given models, at time t=10, how much cooler are the cookies than the hot chocolate? To determine how much cooler the cookies are than the hot chocolate. The cookies are cooler than the hot chocolate

11. Newton and Leibniz

12. Pre-Calculus Integration Integration can be traced as far back as ancient Egypt, 1800 BC, with the Moscow Mathematical Papyrus demonstrating knowledge of a formula for the volume of a pyramidal frustum. The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus, which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation [to the area of a circle. Similar methods were independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere. That same century, the Indian mathematician Aryabhata used a similar method in order to find the volume of a cube. The next major step in integral calculus came from the Abbbasid Caliphate when the 11th century mathematician Ibn al-Haytham devised what is known as “Alhazen’s problem”, which leads to an equation of the fourth degree, in his Book of Optics. While solving this problem, he performed an integration in order to find the volume of a parabolic. Using mathematical induction, he was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree. The next significant advances in integral calculus did not begin to appear until the 16th century. At this time the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of x^n up to degree n=9 in Cavalieri’s quadrature formula. Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri’s method, computing integrals of x to a general power, including negative powers and fractional powers. At around the same time, there was also a great deal of work being done by Japanese mathematicians, particularly by Seki Kowa. He made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of exhaustion.

13. Newton and Leibniz The major advance in integration came in the 17th century with the independent discovery of fundamental theorem of calculus by Newton and Leibniz. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.

14. Formalizing integrals While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them “ghosts of departed quantities”. Calculus acquired a firmer footing with the development of limits and was given a suitable foundation by Cauchy in the first half of the 19th century. Integration was first rigorously formalized, using limits, by Riemann. Lebesgue formulated a different definition of integral, founded in measure theory.

15. Notations Isaac Newtown used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with x’, which Newtown used to indicate differentiation. The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675. He adapted the integral symbol,, from an elongated letter s, standing for summa (Latin for “sum” or “total”).

16. References College Board MsZhao.com Math Type Wikipedia