Volumes of solids

Find the volume of the following solids.

The region bounded by y = x/(x + 1), the x-axis, and x = 4 is revolved about the x-axis.

Problem 43E

Volume of solids . Find the volume of the following solids.

The region bounded by y = , the x -axis , and x=4 is revolved about the x-axis.

Answer;

Step 1

In this problem we need to find the volume of the solid founded by the region bounded by y = , the x -axis , and x =4 is revolved about the x-axis.

In order to find the volume, we will be using the following condition.

If f is a function such that for all in the interval , then the volume of the solid generated by revolving, around the x axis, the region bounded by the graph of , the x axis (y = 0) and the vertical lines andis given by the integral

Volume

[ The radius for our cylinder would be the function f(x) and the height of our cylinder would be the distance of each disk ;dx

The volume of each slice would be

V = dx

Adding the volumes of the disks with infinitely small dx would give us to the formula

V = dx

Here , f(x) = radius

dx = height of each disk

[a , b] = total height of a cylinder . ]

Proper fraction definition ; In a rational fraction , if the degree of f(x) < the degree of g(x) , then the rational fraction is called a proper fraction.

The sum of two proper fractions is a proper fraction.

Example;

Partial fractions Depending upon the nature of factors of Denominator ;

When the denominator has repeated linear factors

A repeated linear factor of denominator corresponds partial fractions of the form ;

= + ++......................+

Where A, B, C,............N are constants which can be determined by equating the numerator of L.H.Sto the numerator of R.H.S , and substituting x = a, we get N.

Example;

Improper fraction definition ; In a rational fraction , if the degree of f(x) the degree of g(x) , then the rational fraction is called a improper fraction.

If an improper rational fraction is given for splitting into partial fractions, we first divide f(x) with g(x) till we obtain a remainder R(x) of lower degree than g(x).

First we express the fraction in the form = quotient

Then we resolve the final proper fractions into partial fractions.