Application of Integral

Done by :

Fatma Al-nuaimi

Kashaf Bakali

Parisa Yazdjerdi

Heba Hammud

Nadine Bleibel

Definition of Area by Integral

by : Fatma Al-nuaimi(201004421)

Finding areas by integration.

Using Riemann sum.

Use Riemann sum to find the value of:

The graph:

Step 1:

We can determine the value by subdividing the region into rectangle:

When the number of rectangles n

The area of rectangle is A=L*W

Width=W= 1/n

Length= 1+(1/n)i

So,

i

Step 3: Performing some algebraic manipulation:

i

i

Step 4:

Taking the limit to calculate the area:

Area under the curveBy: Kashaf Bakali

(201105803)

Find the Area enclosed by the parabola and above the x-axis.

As the area to be calculated should be above x-axis so, .

We first find the points of intersection by solving both equations simultaneously. i.e.

So,

Hence, the intersection points are; .

Now, we sketch the graph.

The graph tells us the limit.

In this case, we have to find area from -1 till 3.

So, dx

)

=

Similarly,

Find the area bounded between and

Firstly, we would find the intersection points by solving both given equations simultaneously. i.e.

We get the intersections points as,

We sketch the graph.

Now we have, . We use the formula, In this case,

and. Hence,

Area between two curveBy: Parisa Yazdjerdi

(201005599)

Areas between two curves

Process of finding area between two curves consist of 3 main steps :

1. finding intersection of the curves ( put two equation in an equality)

2. Drawing the graph to distinguish intervals and exact areas

3. Using integral formula

parisa yazdjerdi

Area between two curves

Example : find area between f(x) = Sin x , g(x) = Cos x , x = 0 and x = π/2 .

First step : find intersections

parisa yazdjerdi

Area between tow curves

Second step : Drawing the Graph

parisa yazdjerdi

Area between two curves

Third step : Using formula to find area

parisa yazdjerdi

Area between two curves

parisa yazdjerdi

22222

4

)]2

1

2

1()10[()]10()

2

1

2

1[(

]sincos[]cos[sin

]cos[sin]sin[cos

so,

cosx -sinx|=cosx-sinx| so, 0>cosx -sinx

and

sinx-cosx|=sinx-cosx|so, 0 >sinx-cosx therefore24

cossin4

0;sincos

2

4

40

2

4

4

0

xxxx

dxxxdxxxA

xxxandxxx

Volume of SolidBy : Hebba Hammud

(201003247)

Volume of Solids

For any Solid(S),we cut it into pieces and approximate each piece by a cylinder. This is called : cross-sectional area.

Definition of Volume

)(

)()(lim1

abAAdx

dxxAxxiAv

b

a

b

a

n

in

Exercises(about the x-axis)

Find the volume v resulting from the revolution of the region bounded by:y=√x , from x=0 to x=1 about the x-axis.

2)0

2

1(

2

)(

10

2

1

0

1

0

2

x

xdx

dxxv

Find the volume v resulting from the revolution of the region bounded by:y=√(a2-x2 ) from x=-a to x=a and the x-axis about the

x-axis.

3

4]

3

2[2

]0)3

[(2

)3

(2

)(2

)(

)(

33

32

0

32

0

22

22

222

aa

aaa

xxa

dxxa

dxxa

dxxav

a

a

a

a

a

a

Exercises (about the y-axis)

Find the volume of the solid of revolution generated by rotating the curve y = x3 between y = 0 and y = 4 about the y-axis.

We first must express x in terms of y, so that we can apply the formula.

If y = x3 then x = y1/3

The formula requires x2, so x2 = y2/3

Find the volume generated by the areas bounded by the Given curves if they are revolved about the y-

axis: y2 = x, y = 4 and x = 0 [revolved about the y-axis]

VolumeBy: Nadine Bleibel

(201104593)

Basics of a Cylinder

Nadine Bleibel

A cylinder is a simple solid which is boundedby a plane region B1- which is called the base.A cylinder also has a congruent region B2 in a Parallel plane.

The formula for volume for a circular cylinder isV=(Pi)r^2(h)

EXAMPLE 1 (example 2 p 356)

Nadine Bleibel2

)02

1(

2

)(

10

2

1

0

1

0

2

x

xdx

dxxv

Find the Volume of the solid obtained by rotatingAbout the x-axis the region under the curvey= √x from 0 to 1.

EXAMPLE 2 (example 3 p357)

Nadine Bleibel

EXAMPLE: Find the volume of the solid obtaining by rotating about the y-axis the region bounded by y = x3, y = 8, and x = 0.