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Lecture

# lect118_1_w13.pdf

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Department
Mathematics
Course Code
MATH 118
Professor
Robert Andre

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Monday, January 7 − Lecture 1: Integration by substitution (Refers to 6.1 in your text) After having practiced using the concepts of this lecture the student should be able to: define the differential of a function, integrate indefinite integrals by making appropriate change of variables (substitution), integrate definite integrals by appropriate change of variables. Summary of what we have learned about the notion of integration. - Our study of integration began with attempts at finding the area of the region bounded by the curve of f(x) and the x-axis over an interval [ a, b]. To do this we introduced the notion of a Riemann sum. But computing areas in this way is inefficient. - The Fundamental theorem of calculus presented an alternate way to compute such numbers. This important theorem is presented into two parts. - The second part of the Fundamental theorem of calculus says that to find the area of the region bounded by the curve of f(x) and the x-axis over an interval [ a, b] it suffices to find an anti-derivative of the function f(x), evaluating it at the limits of integration and subtracting the results. That is, if f(x) is continuous on [a, b] and F(x) is an anti- derivative of f(x) then - The first part of the Fundamental theorem of calculus says that, if f(x) is continuous on [a, b] and as x ranges over [a, b], then This can also be expressed by - The first part of the FTC appears to be unrelated to our initial inquiry. In these notes we used it to provide an easier proof of the second part of the FTC. But it is worth remembering it since we occasionally invoke it in particular situations. It is a more difficult statement to grasp, and so in order for it to become familiar to you, it is a good idea to try to use it whenever we can. - Then we introduced some notation. We defined the indefinite integral of a function f (x), or general anti-derivative of a function f (x) to be the family of all functions whose derivative is f (x). The indefinite integral of a function f is denoted as: - Many notions in the study of calculus may appear abstract and unfamiliar to students at first. But students are encouraged to be bold and attempt to apply these in the best way they understand these through worked out exercises. Of course, their solutions will often turn out to be incorrect; but by analyzing each incorrect answer the student’s perception eventually converges towards a proper understanding of these concepts. The study and learning of mathematics is all about “first getting it wrong, once, twice and then getting it right”. Initially, we all interpret mathematical concepts in crude, often incorrect, ways; then we seek to improve on these. - The FTC tells us that to solve definite integrals we should focus our energy, not on finding limits of Riemann sums, but preferably on developing efficient algorithms that can help us find antiderivatives of a function. - We first must review briefly what the differential of a function is. 1.1 Definition − If y = f(x), we define the differential of f(x), df or dy, as being for some number x . (Note: In dy = 1 ′(x)(xf ′(x)∆x the term which is the argument of f′ is the term which is subtracted in ∆x.) 1 2 1.1.1 Example − Suppose y = x + x. Then dy = (2x + 1)∆x. Observe that the value of dy depends on the value of x and the length of the ∆x-interval. We could also write 2 dy = d(x + x) = (2x + 1)∆x. In this example, if x = 3 and ∆x = 2 then dy = 14. 1.1.1.1 Observation − Suppose we consider the function y = x.  Then, by definition, the differential of x, d(x), is dy = d(x) =
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