MAT 21B Lecture Notes - Lecture 2: General Idea, Antiderivative, Shill
§5.1 Area and Estimating with Finite Sums
Differential Calculus &Integral Calculus
•Differential Calculus (Math21A): This part comes from the study of: velocity problem, slope problem,
rate of change etc.
•Integral Calculus (Math21B): This part comes from the study of area problem.
•These two kinds of problems are closely related by a very important theorem: The Fundamental Theorem
of Calculus!
Area problem
Given a domain, how to calculate its area?
For simple domains, we may have geometric formulas: Rectangle, Triangle, Circle, etc.
How about more complicated domains? Typically, there is no hope to find a simple geometric formula.
Idea: Divide very complicated region into unions of not very complicated regions (if possible). Each smaller
piece is a region “under” the graph of a positive function. So, how to calculate the area under the graph of a
positive function?
Area under the graph of a positive function f(x)on [a, b]
General Idea: Approximate complicated “unknown” objects with simple “known” objects, and then take
the limit!
•Approximate the domain by finite unions of rectangles.
•Approximate the area of the domain by the sum of areas of these rectangles
•Take a limit (by letting number of rectangles approaches infinity).
Approximate the region Rby rectangles
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MAT 21B Full Course Notes
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Document Summary
5. 1 area and estimating with finite sums. For simple domains, we may have geometric formulas: rectangle, triangle, circle, etc. Typically, there is no hope to nd a simple geometric formula. Idea: divide very complicated region into unions of not very complicated regions (if possible). Each smaller piece is a region under the graph of a positive function. Area under the graph of a positive function f (x) on [a, b] Typically, we pick a number n, and subdivide the interval [a, b] into n subintervals of equal width (also called length) x = (b a)/n. [a, b]: a = x0 < x1 < x2 < < xi < xi+1 < < xn 1 < xn = b. Note for each i, xi = a + i x = a + (b a) i/n. When x is small, each of these small pieces looks like a rectangle.
