Monday, January 7 − Lecture 1: Integration by substitution (Refers to 6.1 in your
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
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
dy = d(x + x) = (2x + 1)∆x.
In this example, if x = 3 and ∆x = 2 then dy = 14.
220.127.116.11 Observation − Suppose we consider the function y = x.
Then, by definition, the differential of x, d(x), is dy = d(x) =