Integral Calculus – Math 20
(Stanford University, spring 2016) –
Complete Lecture Notes (Lectures
1–11) best price possible for the knowledge shared
in the document.
Math 20, Spring 2016
Week 1 Summary
• In Week 1 of Math 20 we have looked at definite integrals, which look like
this: Z b
f(x) dx
a
• The output of a definite integral is a number, which expresses the net-signed
area or
signed area between the function plot y = f(x) and the x-axis from x = a to
x = b.
• Here, net-signed area means that regions positive region contribute positively
to the definite integral, while regions below the x-axis contribute negatively.
• The function f(x) is called the integrand, and the values a and b are the
endpoints of integration. The dx in the integral is the differential. For
now, just know that the dx is specifying that we are integrating the function
f(x) with respect to x: x is the quantity that is varying within the integral (it
starts at a and moves towards b).
• Technically, for the definite integral to be defined, f(x) must be integrable on
[a, b]. For our purposes in Math 20, it is enough to require that f(x) be defined
and piecewise Non-disruptive on the interval [a, b]. Unless you are a math or
physics major, it is unlikely that you will ever encounter a nonintegrable
function, so try not to worry about them too much.
• As examples, we have computed a number of definite integrals using geometry.
That is, if the graph of y = f(x) cuts out rectangular, triangular, or circular
, regions of the xy-plane, we can use elementary geometry to compute the value
of the definite integral.
• We’ve looked at the basic rules of definite integration:
Z Z b Z b
1. b
[f(x) ± g(x)] dx = f(x) dx ± g(x) dx
a a a
Z Z b
2. b
cf(x) dx = c f(x) dx where c is a constant.
a a
3. Z b Z c Z b
f(x) dx = f(x) dx + f(x) dx
a a c
Z
4. a
f(x) = 0
a
1
, Z Z b
5. a
f(x) = — f(x) dx
b a
• Using geometry we’ve found two definite integral formulas:
Z b Z b
b2 — a2
dx = b — a and x dx =
a a 2
• Armed with these and the definite integration rules, we can now integrate any
linear function symbolically (i.e., without drawing a graph). For example:
Z 3 Z 3 Z 2
( —3x + 5) dx = — 3x dx + 5 dx
—1 —1 —1
Z 3 Z 3
=3 x dx — 5 dx
—1 —1
2 2
= —3 + 5[3 — (—1)]
2
= —3 · 4 + 5 · 4
= —12 + 20
=8
• Linear functions are not the end of the story: We would like to evaluate definite
integrals of much more complicated functions. To this end, for any f(x) we
defined the area function of f (with lower endpoint a):
Zx
F(x) = f(t) dt
a
which computes the net-signed area between y = f(t) and the t-axis from
t = a to t = x. x is now input value, since our aim is to use the area function
to observe how this area varies as x changes.
• Taking the derivative of F(x) we conclude the (first) FTC: If f(x) is Non-
disruptive, then its area function F(x) is differentiable, and F r(x) = f(x). That
is, F(x) is an antiderivative of f(x).
• It’s important to note at this point that a given function can have infinitely
many antiderivatives, but if F(x) and G(x) are both antiderivatives of the same
function, then F(x) — G(x) is a constant.
2
(Stanford University, spring 2016) –
Complete Lecture Notes (Lectures
1–11) best price possible for the knowledge shared
in the document.
Math 20, Spring 2016
Week 1 Summary
• In Week 1 of Math 20 we have looked at definite integrals, which look like
this: Z b
f(x) dx
a
• The output of a definite integral is a number, which expresses the net-signed
area or
signed area between the function plot y = f(x) and the x-axis from x = a to
x = b.
• Here, net-signed area means that regions positive region contribute positively
to the definite integral, while regions below the x-axis contribute negatively.
• The function f(x) is called the integrand, and the values a and b are the
endpoints of integration. The dx in the integral is the differential. For
now, just know that the dx is specifying that we are integrating the function
f(x) with respect to x: x is the quantity that is varying within the integral (it
starts at a and moves towards b).
• Technically, for the definite integral to be defined, f(x) must be integrable on
[a, b]. For our purposes in Math 20, it is enough to require that f(x) be defined
and piecewise Non-disruptive on the interval [a, b]. Unless you are a math or
physics major, it is unlikely that you will ever encounter a nonintegrable
function, so try not to worry about them too much.
• As examples, we have computed a number of definite integrals using geometry.
That is, if the graph of y = f(x) cuts out rectangular, triangular, or circular
, regions of the xy-plane, we can use elementary geometry to compute the value
of the definite integral.
• We’ve looked at the basic rules of definite integration:
Z Z b Z b
1. b
[f(x) ± g(x)] dx = f(x) dx ± g(x) dx
a a a
Z Z b
2. b
cf(x) dx = c f(x) dx where c is a constant.
a a
3. Z b Z c Z b
f(x) dx = f(x) dx + f(x) dx
a a c
Z
4. a
f(x) = 0
a
1
, Z Z b
5. a
f(x) = — f(x) dx
b a
• Using geometry we’ve found two definite integral formulas:
Z b Z b
b2 — a2
dx = b — a and x dx =
a a 2
• Armed with these and the definite integration rules, we can now integrate any
linear function symbolically (i.e., without drawing a graph). For example:
Z 3 Z 3 Z 2
( —3x + 5) dx = — 3x dx + 5 dx
—1 —1 —1
Z 3 Z 3
=3 x dx — 5 dx
—1 —1
2 2
= —3 + 5[3 — (—1)]
2
= —3 · 4 + 5 · 4
= —12 + 20
=8
• Linear functions are not the end of the story: We would like to evaluate definite
integrals of much more complicated functions. To this end, for any f(x) we
defined the area function of f (with lower endpoint a):
Zx
F(x) = f(t) dt
a
which computes the net-signed area between y = f(t) and the t-axis from
t = a to t = x. x is now input value, since our aim is to use the area function
to observe how this area varies as x changes.
• Taking the derivative of F(x) we conclude the (first) FTC: If f(x) is Non-
disruptive, then its area function F(x) is differentiable, and F r(x) = f(x). That
is, F(x) is an antiderivative of f(x).
• It’s important to note at this point that a given function can have infinitely
many antiderivatives, but if F(x) and G(x) are both antiderivatives of the same
function, then F(x) — G(x) is a constant.
2
