Transcript Applications of Integration

Application of Definite
Integrals
Dr. Farhana Shaheen
Assistant Professor
YUC- Women Campus
Calculus (Latin, calculus, a small
stone used for counting)
 Calculus is a branch of mathematics with
applications in just about all areas of science,
including physics, chemistry, biology, sociology
and economics. Calculus was invented in the
17th century independently by two of the
greatest mathematicians who ever lived, the
English physicist and mathematician Sir Isaac
Newton and the German mathematician
Gottfried Leibniz.
 Calculus allows us to perform calculations that
would be practically impossible without it.
Calculus is the study of change
 Calculus is a discipline in mathematics focused
on limits, functions, derivatives, integrals, and
infinite series. Calculus is the study of change, in
the same way that geometry is the study of
shape and algebra is the study of operations
and their application to solving equations.
 This subject constitutes a major part of modern
mathematics education. It has widespread
applications in science, economics, and
engineering and can solve many problems for
which algebra alone is insufficient.
 Calculus is a very versatile and valuable
tool. It is a form of mathematics which was
developed from algebra and geometry. It is
made up of two interconnected topics:
 i) Differential calculus
 ii) Integral calculus.
 Differential calculus is the mathematics of
motion and change.
 Integral calculus covers the accumulation of
quantities, such as areas under a curve or
volumes between two curves.
 Integrals and derivatives are the basic tools of
calculus, with numerous applications in science
and engineering. The two ideas work inversely
together in Calculus.
 We will discuss about integration and its
applications.
INTEGRATION
 Integration is an important concept in
Mathematics and, together with differentiation, is
one of the two main operations in Calculus.
 A rigorous mathematical definition of the integral
was given by Bernhard Riemann.
Definite and Indefinite Integrals
 Integration may be introduced as a means of
finding areas using summation and limits. This
process gives rise to the definite integral of a
function.
 Integration may also be regarded as the reverse of
differentiation, so a table of derivatives can be read
backwards as a table of anti-derivatives. The final result
for an indefinite integral must, however, include an
arbitrary constant, because there is a family of curves
having same derivatives, i.e. same slope.
Indefinite Integrals
 The term Indefinite integral is referred to
the notion of antiderivative, a function F
whose derivative is the given function ƒ. In
this case it is called an indefinite integral.
Some authors maintain a distinction
between anti-derivatives and indefinite
integrals.
Indefinite Integral as Net Change
 Problem 1:
A particle moves along the x-axis so that its acceleration
at any time t is given by a(t) = 6t - 18. At time t = 0 the
velocity of the particle is v(0) = 24, and at time t = 1 its
position is x(1) = 20.
(a) Write an expression for the velocity v(t) of the
particle at any time t.
(b) For what values of t is the particle at rest?
(c) Write an expression for the position x(t) of the
particle at any time t.
(d) Find the total distance traveled by the particle
from t = 1 to t = 3.
Solution:

(a) v(t) = ∫ a(t) dt = ∫ (6t - 18) dt = 3t2 - 18t + C
24 = 3(0)2 - 18(0) + C
24 = C
2
so v(t) = 3t - 18t + 24
 (b) The particle is at rest when v(t) = 0.
3t2 - 18t + 24 = 0
t2 - 6t + 8 = 0
(t - 4)(t - 2) = 0
t = 4, 2
(c) x(t) = ∫ v(t) dt = ∫ (3t2 - 18t + 24) dt = t3 - 9t2 + 24t + C
20 = 13 - 9(1)2 + 24(1) + C
20 = 1 - 9 + 24 + C
20 = 16 + C
4=C
so x(t) = t3 - 9t2 + 24t + 4
Calculating the Area of Any
Shape
 Although we do have standard methods to
calculate the area of some known shapes,
like squares, rectangles, and circles, but
Calculus allows us to do much more.
Trying to find the area of shapes like this would
be very difficult if it wasn’t for calculus.
Riemann integral
 The Riemann integral is defined in terms of Riemann
sums of functions with respect to tagged partitions of an
interval. Let [a,b] be a closed interval of the real line;
then a tagged partition of [a,b] is a finite sequence
 This partitions the interval [a,b] into n sub-intervals [xi−1,
xi] indexed by i, each of which is "tagged" with a
distinguished point ti ε [xi−1, xi]. A Riemann sum of a
function f with respect to such a tagged partition is
defined as
Riemann integral
 Approximations to integral of √x from 0 to
1, with ■ 5 right samples (above) and ■ 12
left samples (below)
Riemann sums
 Riemann sums converging as intervals
halve, whether sampled at ■ right,
■ minimum, ■ maximum, or ■ left.
Definite integral
 Given a function ƒ of a real variable x and
an interval [a, b] of the real line, the
definite integral
is defined informally to be the net signed
area of the region in the xy-plane bounded
by the graph of ƒ, the x-axis, and the
vertical lines x = a and x = b.
Measuring the area under a curve
 Definite Integration can be thought of as
measuring the area under a curve, defined
by f(x), between two points (here a and b).
Definite integral of a function
 A definite integral of a function can be
represented as the signed area of the
region bounded by its graph.
Definite integral of a function
 The principles of integration were
formulated independently by Isaac Newton
and Gottfried Leibniz in the late 17th
century. Through the fundamental theorem
of calculus, which they independently
developed, integration is connected with
differentiation as:
Fundamental Theorem of Calculus
 Let f(x) be a continuous function in the
given interval [a, b], and F is any antiderivative of f on [a, b], then
Area between two curves y = f(x)
and y = g(x)
DEFINITION
 If f and g are continuous and f (x) ≥ g(x) for
a ≤ x ≤ b, then the area of the region R
between f(x) and g(x) from a to b is
defined as
b


f
(
x
)

g
(
x
)
dx

a
Area between two curves y = f(x)
and y = g(x)
Examples
Definite Integrals to find the
Volumes
 We can also use definite integrals to find
the volumes of regions obtained by
rotating an area about the x or y axis.
Solid of Revolution
 A solid that is obtained by rotating a plane
figure in space about an axis coplanar to
the figure. The axis may not intersect the
figure.
 Example:
Region bounded
between y = 0,
y = sin(x), x = π/2,
x = π.
Volumes by slicing
I- Disk Method
 A technique for finding the volume of a solid of
revolution. This method is a specific case of volume by
parallel cross-sections.
II- Washer Method
 Another technique used to finding the volume of a solid
of revolution. The washer method is a generalized
version of the disk method.
 Both the washer and disk methods are specific cases of
volume by parallel cross-sections.
Volumes by Disk method
 Let S be a solid bounded by two parallel
planes perpendicular to the x-axis at x = a
and x = b. If, for each x in [a, b], the crosssectional area of S perpendicular to the
x-axis is A(x) = π(f(x))2 ,then the volume of
the solid is
Example: Right circular cone
Find the volume of
the region bounded
between y = 0,
x = 0, y = -2x+3.
Here r = 3.
So, f(x) = -2x + 3 in
the interval [0, 3]
Example: To find volume of a Solid
of Revolution
Problem
Solution
Solid of revolution for the function
2
y
2
1  ( x  2)
Revolution about the y- axis:
d
V
d
 A( y) dy  


f
(
y
)

c
c
2
dy
For f(x) = 2 + Sin x, revolved about
the x – axis
The volume is
2
V
  (2  Sin x)
0
2
dx
Solids of Revolution
Region bounded between
y = 0, x = 0, y = 1,
y = x2 +1.
Objects obtained as Solids of
revolution
Volumes by washer method
perpendicular to the x-axis
 The volume of the solid generated when
the region R, (bounded above by y=f(x)
and below by y=g(x)), is revolved about
the x-axis, is given by
Washers
(disk with a circular hole)
Washer shapes
in everyday life
Volume by slicing-washer method
Volumes by washer method
perpendicular to the y-axis
 The volume of the solid generated when
the region R (bounded above by x=f(y)
and below by x=g(y)), is revolved about
the y-axis, is given by
Volumes by washer method
 Figure illustrates how a washer can be
generated from a disk. We begin with a disk
with radius rout and thickness h. A smaller
concentric disk with radius rin is removed from
the original disk. The resulting solid is a washer.
The Washer Method for Solids of
Revolution
Volume of region bounded
between
, y = x 2.
 When the solid is formed by revolving the
region between the graphs of
y = f(x) and y = g(x), where f(x) > g(x),
about the y-axis, the height of the
rectangle is given by h = f(x)-g(x).
Volumes by cylindrical shells
 A cylindrical shell is a solid enclosed by
two concentric right circular cylinders.
 Let f be continuous and non-negative on
[a, b], 0≤a≤b , and let R be the region that
is bounded above by y = f(x), below by the
x-axis, and on the sides by the lines x = a
and x = b. Then the volume of the solid of
revolution that is generated by revolving
the region R about the y – axis is given by
Volume by cylindrical shells about
the y-axis
Method of shells
 The method of shells is fundamentally
different from the method of disks. The
method of disks involves slicing the solid
perpendicular to the axis of revolution to
obtain the approximating
elements. However, the method of shells
fills the solid with cylindrical shells in which
the axis of the cylinder is parallel to the
axis of revolution.
 To illustrate the computation of the volume
of a cylindrical shell, a paper towel roll or
toilet paper roll can be an example too.
 Central to the development of the method
of shells is the idea of nesting or layering
of the approximating elements. The notion
of nesting can be introduced using the
layers of an onion.
Shells made by Russian dolls
 Another useful prop to illustrate this idea is
a set of Matroyska dolls. In the Figure
below, we see that the hollow dolls of
varying sizes nest together compactly.
Region bounded between
y = 0, y = sin(x), x = 0,
x = π.
 The animation in the next Figure illustrates the
steps involved with the shell method for
computing the volume of the solid of revolution
generated by revolving the region in the first
quadrant bounded by the graph of y = sin(x) and
the x-axis about the y-axis. First, the region is
partitioned and a typical shell is
drawn. Approximating half-shells are drawn. To
complete the visualization, the approximating
shells are produced. After the approximating
shells are drawn, the solid of revolution is
generated.
Solids of Revolution: The
Method of Shells
Region bounded
between y = 0,
y = -12+8x-x2,
x = 2, x = 6.
Region bounded
between y = 0,
y = sin(x), x = π/2,
x=π
Partition/Shell
Shells
For method of shells…
 We focus on regions bounded by the
graph of a continuous function y = f(x) on
the interval [a,b], the vertical lines x = a
and x = b. For the illustrations we also
require that f(x) is nonnegative over
[a,b]. Several regions of this type are
shown in the Figure.
THANKYOU