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Application of Definite Integrals

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Title: Application of Definite Integrals


1
Application of Definite Integrals
  • Dr. Farhana Shaheen
  • Assistant Professor
  • YUC- Women Campus

2
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.

3
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.

4
  • 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.

5
  • 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.

6
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.

7
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.

8
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.

9
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.  

10
Solution
  • (a)  v(t) ? a(t) dt ? (6t - 18) dt 3t2 -
    18t C                          24 3(0)2 -
    18(0) C                          24 C      
    so  v(t) 3t2 - 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

11
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 wasnt for
    calculus.

12
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 e
    xi-1, xi. A Riemann sum of a function f with
    respect to such a tagged partition is defined as

13
Riemann integral
  • Approximations to integral of vx from 0 to 1,
    with  5 right samples (above) and  12 left
    samples (below)

14
Riemann sums
  • Riemann sums converging as intervals halve,
    whether sampled at  right,  minimum,  maximum,
    or  left.

15
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.

16
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).

17
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.

18
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

19
Fundamental Theorem of Calculus
  • Let f(x) be a continuous function in the given
    interval a, b, and F is any anti-derivative of
    f on a, b, then

20
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

21
Area between two curves y f(x) and y
g(x) Examples
22
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.

23
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
p/2, x p.
24
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.

25
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 cross- sectional
    area of S perpendicular to the
  • x-axis is A(x) p(f(x))2 ,then the volume of
    the solid is

26
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Example Right circular cone
Find the volume of the region bounded between
y 0, x 0, y -2x3. Here r 3. So, f(x)
-2x 3 in the interval 0, 3
29
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30
Example To find volume of a Solid of Revolution
  • Problem

31
Solution
32
Solid of revolution for the function
33
Revolution about the  y- axis
34
For f(x) 2 Sin x, revolved about the x axis
  • The
    volume is

35
Solids of Revolution
36
Region bounded between y 0, x 0, y 1,  y
x2 1.
37
Objects obtained as Solids of revolution
38
Volumes by washer method perpendicular to the
x-axis
  • The volume of the solid generated when the region
    R, (bounded above by yf(x) and below by yg(x)),
    is revolved about the x-axis, is given by

39
Washers (disk with a circular hole)
40
Washer shapes in everyday life
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42
Volume by slicing-washer method
43
Volumes by washer method perpendicular to the
y-axis
  • The volume of the solid generated when the region
    R (bounded above by xf(y) and below by xg(y)),
    is revolved about the y-axis, is given by

44
Volumes by washer method
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  • 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.

47
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The Washer Method for Solids of
Revolution Volume of region bounded between , y
x2.
50
  • When the solid is formed by revolving the region
    between the graphs of  y f(x) and y g(x),
    where f(x) gt g(x), about the y-axis, the height
    of the rectangle is given by h f(x)-g(x).

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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,
    0ab , 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

53
Volume by cylindrical shells about the y-axis
54
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.

55
  • To illustrate the computation of the volume of a
    cylindrical shell, a paper towel roll or toilet
    paper roll can be an example too.

56
  • 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. 

57
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.

58
Region bounded between y 0, y sin(x), x
0, x p.
59
  • 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.

60
Solids of Revolution  The Method of Shells
61
Region bounded between y 0, y -128x-x2, x
2, x 6.
62
Region bounded between y 0,  y sin(x), x
p/2, x p Partition/Shell
Shells
63
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.

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