Calc 1A

Assignment Five

A problem which was initially dealt with by the ancient Greeks was to determine (or estimate closely) the area of irregularly shaped regions. The method of 'exhaustion' was used to estimate the area of a circle: regular polygons who's areas could be easily computed were inscribed in a circle, then as the number of sides were increased, it was discovered that the estimate got better and in fact approached a limiting value (what we know today as pi * r squared). It was through this method that the value of pi began to be determined as well, by the study of a circle of radius one.

We have largely finished working with derivatives in applications and hence have completed the study of introductory 'differential calculus'. Now we begin that which is referred to as 'integral calculus' by introducing basic constructions which we can use to estimate areas enclosed between curves. Why we are interested in area under curves is another story (an important and useful one).

The Definite Integral and its notation will be initially introduced as equivalent to an area of a nonnegative function. Soon we will see how the Definite Integral relates to areas for negatively valued functions as well. We will not introduce the process of integration just yet; instead we will define the definite integral in terms of limits of infinite Riemann sums.

Later in the term, we will learn to integrate and by way of the Fundamental Theorem of Calculus, we will see how that act relates to integration. For now we will keep our eye on finite approximation techniques and lead from there to the infinite Riemann sum limits.Many students find the notation throughout to be the most challenging aspect of all this. For that reason, it is important that you become familiar with the notation early-as soon as it is introduced.

Day	Class Topic	Homework-Reading Class notes and Examples is part of EVERY NIGHT'S homework.
One	Area approximation: Finite Riemann sums AND sigma notation/properties, definite integral intro/notation for non-negative functions	Pg 137 #(not a mistake) #13,17,37, 39 Pg 374-375 # 9,11,15  Pg 402 # 1, 3-9 odd, 15-19 odd. Pg 414 #1,3
Two	Right, Left, midpoint sums/inscribed, circumscribed. Infinite Riemann Sums. Limits.	Pg 414-415 #7, 39(this is a regurgitation of what we did in class) and repeat #39 instructions for y=x^2 + 1 on [-1,2] .
Three	More. Practice/Clarification. Calculator as helpful tool. Integrability.	Pg 414-416 #3,9, 41, 43, #15 for 'extra fun'.
Four	Properties of Definite Integrals. Definite Integral vs. area. "Signed Area".	Pg 415-416 #17, 19 (HINT: x^2 +y^2=1, you have seen this function before), 21,23,27a 45a,b
Five	Practice/Clarification	Pg 415-416 #25,27b,29,45c,d,
Six	Approximating Area: Trapezoidal approximation for non-negative functions	Pg 562 #1,3,5 'b' only, use n=5 instead (answers respectively are: 4.659, 1.934, .322 to three decimals). Pg 562 #21 (ignore instructions, do trapezoidally), again use n=5 (answer is 2.133 to 3 places).