 
Hello students, today I am going to start
 
the surface area of solids of revolution
if any curve rotates around the 'x' axis
or rotate around the 'y' axis (2-dimensional curve)
if we rotate the circle around the 'x' axis
it will look like a sphere
how to calculate surface of a sphere
before this, in my earlier videos, I have taught you how to calculate the volume, if you have not seen that video
then click at 'i' tab and
you will get a notification
and you can through that video and check the content I have covered
before this, you have to understand the concept of curve tracing
also, refer to this go and check the video on curve tracing
then you can learn this topic
so today I m going to tell you if 2 dimension curve
rotate around the 'x' axis than how to calculate the surface area
the formula of surface area denoted by 'S'
 
 
 
if it rotates around the 'x' axis
 
if it rotates around the 'y' axis there is a limit
 
 
 
 
 
 
I will take an example and a very simple example
 
it is a circle
if anyone asks you that this circle
rotates around the 'x' axis
and the shape looks like and what will be the surface area,
so, student, this is a circle and rotates around 'x' axis
 
we know what will be it's coordinate
and I told you
in volume chapters, if any 2 dimensional
curve rotates around the 'x' axis
its upper half limit is useful and the lower half limit is not used to us
if we remove the lower half
then also it will look like a sphere
there is no role of the lower half we will take upper half only
 
 
 
 
 
 
 
sometimes students by mistake do 4 times which is not right
we have to do twice only
 
now we will  do differentiate
I will tell you how to do differentiate
 
 
 
 
 
 
now we will take L.C.M
 
 
 
 
 
 
 
 
 
 
 
 
 
 
So in this way, we will solve the question
this is the surface area
so I will take some more question
 
 
 
in my previous video
I have taught you volume, I have covered this question
but there was direct equation was given
here parametric coordinates
from parametric coordinates you have to find out the equation
then you have to trace the curve, the same question
which I have told you in volume video you  can go through there
 
 
 
firstly we will convert in the form of an equation
 
 
find out the value of 't'
 
 
 
 
 
 
 
 
 
we will simplify this
 
if we take out (-) as common than also it will not effect
now what we will do is, loob that is created
that loob is rotating around 'x' axis
 
you can see in volumes video I have taken the same question
 
 
 
 
like this loob is created
 
 
this loob is rotating around 'x' axis  so we have
to calculate the surface area
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
the limit will be form 0 to 3
here we will not take  twice
because only the upper half is the useful, not lower half
if lower half is not there and we rotate  than
the same shape as the upper half will be there
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
value of 'y' square is given
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
like this, we will solve this question
let's take the last question
 
 
 
 
 
 
this I have told you in volume's chapter
you can through and check
how I have calculated volume, revolving around the 'x' axis
and today I will tell you how to calculate surface area
this question is important and based on exam
 
you have  to remember this shape
and it is rotating
around 'x' axis
so you have to calculate the surface area
you should know the parametric coordinate
 
 
parametric coordinate are those coordinates which satisfy this equation
 
and you can find all the coordinate from parametric coordinates
and this generates  the curve
 
 
 
 
 
 
 
you know we will take upper half limit
and we will not take lower half limit
 
 
 
 
 
 
 
now see the differentiation
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
now look at the following steps
 
 
 
 
 
 
 
in last class, I have told you about gamma beta
you can take Sin as 't'
but I will tell you gamma beta, a short trick I have told you
 
 
 
 
those students who don't know I want to tell you  that
 
 
 
 
 
 
 
 
 
 
 
 
this concept
I have used in volumes chapter, you can see there also
now what we will do
in this concept
 
 
 
 
 
 
 
 
 
 
this is the answer like this we will solve
this question, so the student go and watch my videos
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thank you so much for watching my videos
next topic will be double integration
change of order of integration
who we can change the order of double integration
thank you
