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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

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

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.

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

Riemann integral

- Approximations to integral of vx 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 anti-derivative 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

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

p/2, x p.

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 cross- sectional

area of S perpendicular to the - x-axis is A(x) p(f(x))2 ,then the volume of

the solid is

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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

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Example To find volume of a Solid of Revolution

- Problem

Solution

Solid of revolution for the function

Revolution about the y- axis

For f(x) 2 Sin x, revolved about the x axis

- The

volume is

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 yf(x) and below by yg(x)),

is revolved about the x-axis, is given by

Washers (disk with a circular hole)

Washer shapes in everyday life

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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 xf(y) and below by xg(y)),

is revolved about the y-axis, is given by

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.

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The Washer Method for Solids of

Revolution Volume of region bounded between , y

x2.

- 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

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 p.

- 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 -128x-x2, x

2, x 6.

Region bounded between y 0, y sin(x), x

p/2, x p 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.

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THANKYOU