Hi, everyone.
Welcome back to integralcalc.com.
Today, we’re going to be talking about how to find the volume that’s obtained when
you revolve the parametric curve around the x axis.
So this is a volume of revolution problem.
But in this particular problem, we’re dealing with parametric curves.
So we have the parametric curve that’s defined by the two equations x equals t cubed and
y equals 2t squared plus 1.
We’ve been asked to find the volume of revolution when you revolve that parametric curve around
the x axis between t equals negative 1 and t equals positive 1.
So I’ve got several formulas up here.
The first thing we need to know is that whenever we’re dealing with parametric curves, the
equation for x is representative of f of t and y represents g of t.
So those are just the same.
So I wanted to make sure that we remembered that because we’re going to need it.
The formula we’re going to use for volume is this b sub x formula, x meaning we’re
revolving around the x axis.
And this formula here pi y squared inside the integral is specific to revolving around
the x axis.
We already know that a and b are our limits of integration so the only thing we need to
determine whenever you’re dealing with volume of revolution, you need to know which of your
limits of integration produces a larger value when you plug it in to f of t.
So let’s talk about what we need.
So we have these limits of integration, negative 1 and positive 1 and at first glance you would
think no problem, negative one is going to be a, 1 is going to be b and those are going
to be our limits of integration that go into our formula here because that’s how it normally
is.
But when we’re dealing with either area under the parametric curve or volume of revolution
for a parametric curve, you actually have to plug in these limits of integration to
your equation for x or f of t.
So we have x equals t cubed.
We need to plug in negative 1 and 1 for t into that equation and see which value is
larger.
So let’s go ahead and do that really quick.
We’ll get f of negative 1 equals negative 1 cubed which is going to give us negative
1.
If we plugged in positive 1, f of 1 equals 1 cubed and we’ll get a positive 1.
So in this case, our negative 1 limit of integration did end up being less than our positive 1
limit of integration.
So if we say that negative 1 is a and positive 1 is b, then f of a is less than f of b which
means that these are our limits of integration here with a on the bottom and b at the top.
If this had been the opposite such that plugging in negative 1 had produced a larger result
than plugging in positive 1, we would have to integrate from b to a.
we would have to flip our limits of integration and evaluate from 1 to negative 1.
In other words, right to left instead of negative 1 to positive 1 as we would normally.
So let’s go ahead and start building our formula.
Volume of revolution around the x axis is going to be from negative 1 to 1 because we
confirmed that that’s the direction we’re going by plugging in our limits of integration.
pi here is part of the formula that will always be there.
Now we’ve got this y squared so y we’re going to grab from this equation here from
y that was given to us in our problem so we need to plug in 2t squared plus 1 and according
to our formula, we need to square that.
And then we have, here in our formula, dx.
And dx is going to be defined as taking the derivative of x.
So we have this equation, x equals t cubed.
So d of x is going to be the derivative of that equation for x which of course will be
3t squared dt.
So we’re going to be plugging in 3t squared dt for dx.
So now we’ve got our integral set up, everything is in terms of t so we just need to simplify
our integral as much as possible then integrate and evaluate on the limits of integration
and that’ll give us the volume of revolution of this parametric curve around the x axis.
First of all, we can pull the 3 and the pi out in the front of our integral because everything
inside of our integral is multiplied together and those are both constants, they’re both
coefficients so they can come outside the integral so we don’t have to worry about
them.
So now we end up with t squared times 2t squared plus 1 squared dt.
So we want to go ahead and multiply out that 2t squared times 1 squared so we’ll get
t squared times, when we multiply that out, we’ll end up with 4t to the 4th plus 4t
squared plus 1 dt.
Then if we distribute this t squared, we multiply that by everything inside the parenthesis,
we’ll end up with 4t to the 6th plus 4t to the 4th plus t squared dt.
Now, we can easily go ahead and take the integral so volume of revolution around the x axis,
we’ve got 3pi times when we take our integral remember we’re adding 1 to the exponent
then dividing the coefficient by the new exponent so 4t to the 6th.
We add one to the exponent to get t to the 7th.
Then we divide the coefficient by the new exponent,7 and we end up with 4/7 t to the
7th.
Then similarly 4/5 t to the 5 plus 1/3 t cubed.
And we’ll be evaluating that on the range negative 1 to positive 1.
So we can go ahead and plug in our limits of integration.
Remember, you always want to plug in you upper limit of integration first so we’ll end
up with 4/7 times 1.
1 to the 7th power or 1 to any power is just going to be 1.
1 times 4/7 is just going to give us 4/7.
So we’ll get as you can see plus 4/5 plus 1/3.
Then we’ll subtract what we get when we plug in our lower limit of integration.
So negative 1 to the 7th power is going to be negative 1 so we’ll end up with a negative
4/7.
Negative 1 to the 5th power is also 1 times 4/5 is negative 4/5 so minus 4/5.
And negative 1 to the 3rd is also negative 1 so we end up with minus 1/3.
So now, we’re really just down to simplifying here.
We get 3pi times 4/7 plus 4/5 plus 1/3 and then we have a double negative for each one
of these.
So we’ll end up with plus 4/7 plus 4/5 and plus 1/3 and we can combine those fractions
that have common denominators and we’ll get 4/7 plus 4/7 is going to be 8/7.
4/5 plus 4/5 gives us 8/5 and then 1/3 plus 1/3 is 2/3.
So if we want to combine these fractions, we need a common denominator.
If we multiply 7 by 5 by 3 to get a common denominator, we’ll end up with 105.
So we need to multiply 8 times 15 for this first fraction here so basically, we’ll
end up with 120/105 plus multiplying 8 by 21 gives us 168/105 and multiplying 2 by 35
gives us 70/105.
If we combine those, we get 3pi times 120 plus 168 plus 70 is 358/105.
We can cancel 105 with 3.
We’ll end up with just 35 in the denominator.
So our volume of revolution around the x axis is going to be 358pi over 35.
And that’s it.
That’s our final answer.
That is the volume obtained by revolving this particular parametric curve around the x axis
on the range negative 1 to 1.
I hope this video helped you guys and I will see you in the next one.
Bye!