Areas and Volumes By Slices

We can use integrals to locate the area below an arc. In a number of applications we got to locate the un-identified area instead of the signed area. For example, if the areas below the arc represent a physical object and want to identify how much cover to buy to cover it, would similar to identify the sum area. If we got to find the un-identified area among two function from a to b, we can use the following formula,                                      

                                                               `int_a^b|f(x)-g(x)|dx` .
   

 

Volumes by Slices:

 

            We can use integrals to estimate the area below an arc or revolved about an axis. Integrals can also be used to estimate the volumes of objects.  We think how to estimate the volumes of different group of objects. For one group of volumes, contain a formula for the cross sectional area. The differential volume of a cross section is given by

            dV = A(x)dx

The total volume of a cross sections can be written using, 

            V =`int_a^bA(x)dx`

At a point x, the differential volume by this method is

            dV = `pi` y2 dx

Volume through revolution concerning the x axis among a and b is

            y = f(x) - g(x)

For revolution concerning the x axis, we have

            V =`int_a^bpi[f(x)-g(x)]^2dx`

If the volume is leaning concerning the y axis after that instead we have.

            V =`int_a^bpi[s(y)-t(y)]^2dx`

There is also a method intended for an offset axis of revolution. Among a < x < b then write

            V =`int_a^bpi(f(x)-h)^2dx`

If  you wish for to do a volume of revolution by means of cylindrical shells and have concerning an axis y = h after that you would have for x > h.

            V =`int_a^b2pi(x-h)ydx`

     

 

Examples of areas and volumes by slices:

 

Example of areas by slices:

  • Example: 1

In arrange to locate the area below a graph requires to state the lower and upper values of x.

                                       

Locate this area contain to integrate between x = 1 and x = 3.

Solution:

`int_1^3 x^2dx`

`int_1^3 x^2dx`  = `[x^3/3]_1^3`

                = `[3^3/3]`

                = `[27/3]`          

                      = 9 square units

Examples of volumes by slices:

  • Example: 2

Locate the volume of a pyramid through base area B and height h through flat cross sections.

Solution:

We arrange in a line the pyramid upside down along the y axis and at the origin. If get some y cross section of the pyramid then

            A(y) = B(y2 / h2)

The area scales with the square of the height since it is relative to the square of the measurement. We canister estimate the volume of the pyramid through cross sections.

           V = `int_0^hA(x)dy`

            V =`int_0^h``B(y^2/h^2)dy `

            V = Bh / 3

 

  • Example: 3

Locate the volume of a sphere of radius R through circular cross sections.

Solution:

Obtain a circle centered at (0, 0) through radius R. A circular cross section which passes through the sphere can exist established from.

            x^2+y^2=R^2                                                                               

            y=±`sqrt(R^2-x^2)`


                 `int pi y^2 dx = int_-R^R pi (R^2 - x^2) dx = pi(R^2x - ((x^3)/3))| _-R^R`    

We wish to take the top division of y and revolve it around the x axis to find the volume of revolution for a sphere.