Let's derive the area for a sphere
So we know from previously that the circumference of a circle is equal to 2 Pi r?
so if we come up some some arbitrary Angle theta like so
We see that the the radius of the circles become they start it starts off at Capital R
which this is the radius this is the circle that we're going to be deriving the
area for all right well this is the sphere [that] will be driving the area for
But we see that the circle right there is is this big but then it gets smaller and smaller and smaller small so we need?
to develop an equation that relates that relationship to
this smaller circle, and we're going to find the circumference of this smaller circle, and then we're going to
multiply times
Some little element d. S right and
[that] will give us a differential element if we multiply the circumference of that [small] circle times the differential Element D
s will get a little bitty area right here [a] strip all the way around and
If we sum up all the way, you know all along the sphere
Then we will get we will trace out
The area of the [sphere], and then that will give us an equation for the area of the sphere
So we know also from previously that s is equal to R theta?
So taking the differential we say that d s is equal to or d theta
right
so then the area for a sphere
Is equal to the integral?
Summing up all these little areas Da
but then that the area of one of these little rings is
equal to
2 Pi R times [Ds]. Right 2 Pi R
Times D s but then d s is equal to [r]. [d] theta?
all right, and this is 2 Pi or
Yeah, that's correct, and then this is our d theta
Now the next relationship is to find
This radius this smaller radius as a function of theta
well, we see that this the radius of the sphere here is the same as the radius of the sphere here and
Then this angle in here
This angle right here is the same as this angle theta so we can say that this is equal to R times cosine theta
Right so then this is equal to
[2Pi] and replace this all with that aurel cosine theta
or D theta
so then we have the sum and we're going to
Factor out some constants [-] PI this R and this r become r. Squared and they are constants. It's a constant value
They're the radius of the sphere right, so this is 2 Pi
Cap R. Squared the integral of
Cosine theta
D theta
Now we have two ways to do the integration
we can integrate from negative Pi over 2 to [Pi] over 2 or we could just integrate from 0 from an angle of 0
to an angle of Pi over 2 right and
You know from negative Pi [over] 2 be down here and then pI over 2 is up here?
But let's just integrate from 0 to pi over 2 and that will give [us] the [area] of this
hemisphere and then multiply Times 2
So we'll integrate from 0 to PI over 2 and
then multiply that whole thing Times 2
So I'm going to factor this - all the way over 4 Pi
R squared
0 to Pi over 2
Cosine Theta [d-Theta]
The antiderivative of Cosine theta is sine theta
evaluated from [Zero] to Pi over two
like so
Sine of Pi over [two] is [one]
Sine of Zero is zero
so this thing this entire thing becomes one and
One minus zero so then we have the area of the sphere is equal to 4 pi times cap [R] squared
So this is the equation for the area of a [sphere]
Okay, let me step the [side] of your clear screenshot
