We want to find the arc length of the spiral
given by the parametric equations for t
on the closed interval from zero to two.
Let's begin by looking at this graphically.
The graph of the spiral looks like this,
and when t equals zero, we'd be at this point on the spiral,
and when t equals two, we'd be at this point on the spiral.
So the distance from this point to this point
along the curve or the length of this piece of the spiral
is the arc length we're trying to find.
So again, if we start at this point
and travel to the second point along the curve,
the distance traveled is equal to the arc length
we're trying to find.
The arc length is equal to the integral of the square root
of the quantity x prime of t squared,
plus y prime of t squared,
integrated from a to b, or in our case, from zero to two.
Let's begin by determining x prime of t and y prime of t.
Notice how both of these functions are products
of two functions, so we'll have to apply the product rule.
So x prime of t is equal to the first function,
e to the two t, times the derivative of the second function,
the derivative of cosine t is negative sine t,
plus the second function, cosine t times the derivative
of the first function, or e to the two t,
that would be e to the two t, times a derivative of two t,
which is two.
So simplifying, we have negative e to the two t,
sine t,
plus two e to the two t, cosine t.
Again, this is x prime of t,
which you'll be using in the arc length formula,
after we find y prime of t.
So, y prime of t also requires a product rule.
So we've got the first function, e to the two t,
times a derivative of the second function,
the derivative of sine t is cosine t,
plus the second function sine t,
times a derivative of the first function,
or the derivative of e to the two t,
which is e to the two t times two.
Again, simplifying, we have y prime of t
equals e to the two t cosine t,
plus two e to the two t sine t.
And now we can set up the integral
to determine the arc length.
The arc length s is equal to the integral of the square root
of, again, x prime of t squared.
So we'd have negative e to the two t sine t,
plus two e to the two t cosine t.
This is all squared, plus y prime of t squared,
which is e to the two t cosine t,
plus two e to the two t sine t.
And again, this is all squared.
And we're integrating on the integral from zero to two.
Let's begin simplifying the radicand on the next slide.
Let's begin by squaring this, squaring this,
and simplifying.
So, there's no shortcut to square this.
We have to write out the factor twice.
Notice how we'll have four products,
one, two,
three, and four.
Notice for the first product, we'd have a negative times
a negative, that's positive.
And then we'd have e to the four t.
When multiplying and the bases are the same,
we add the exponents, and then we have sine squared t.
For the second product, we would have negative or minus
two e to the four t, sine t times cosine t.
For the third product, we would have negative or minus
two e to the four t, again, sine t times cosine t.
And then finally, for the fourth product,
we would have plus four e to the four t,
cosine squared t.
Notice how these two middle terms are like terms.
So we have the arc length s equals the integral
from zero to two of the square root
of, again, this square would be e to the four t,
sine squared t.
And then we have minus four e to the four t,
sine t cosine t,
and then we have plus four e to the four t,
cosine squared t.
And now from here, we have to square this,
and then finally, simplify the radicand.
So to square this, we'll have these two factors,
and again, we'll have four products,
one, two,
three, and four.
To save some time, I'm going to quickly work this out.
Just as before, these two middle terms are like terms,
so you can combine those to three terms.
So we'll add these three terms under our square root.
So we'd have plus e to the four t, cosine squared t,
plus four e to the four t, sine t cosine t.
And then plus four e to the four t, sine squared t.
From here, notice how these two terms are opposites,
and therefore, they sum to zero.
And then from here, notice that these two terms
share a common factor of e raised to the power of four t.
So let's go ahead and write those in factored form.
If we factor out e to the four t, just from those two terms,
we'd have e to the power of four t times sine squared t,
plus cosine squared t.
And then the remaining two terms, these two terms here,
share a common factor of four e to the four t,
so we'll factor out four e to the four t,
just from these two terms.
So we'd have plus four e to the four t,
times the quantity cosine squared t,
plus sine squared t.
And we should recognize the identity here,
sine squared t plus cosine squared t simplifies to one,
and of course, so does cosine squared t plus sine squared t.
So we have the arc length s equals the integral
from zero to two of the square root of,
this would just be one e to the four t,
plus four e to the four t,
which would be five e to the four t,
which, of course, we can write as
integral from zero to two
of the square root of five,
times the square root of e to the four t.
Let's factor the square root of five out.
So we'd have the square root of five
times the integral from zero to two
of the square root of e raised to the power of four t.
Remember, e to the four t is equal to e to the two t,
times e to the two t.
So the square root of e to the four t
is just e to the two t.
So we have the integral of e to the two t
integrated with respect to t,
again, from zero to two.
Let's go ahead and finish this on the next slide.
Here we'll have to perform u substitution,
where u is equal to two t,
differential equals two dt.
So, dividing both sides by two, we have 1/2 du equals dt.
So we'll have an extra factor of 1/2 we're integrating here
to write this in terms of u.
We could write this as the square root of five
times the integral of e to the u,
but again, dt is equal to 1/2 du.
So we can put the du here and factor out the 1/2.
If we want to leave this in terms of u,
we'd have to change the limits of integration.
Let's go ahead and do that.
Notice when t is zero, u is two times zero over zero.
And when t is two, u is equal to two times two over four.
So we would have the square root of five divided by two,
times e to the u.
And again, the limits of integration are from zero to four.
So we have square root five divided by two,
times when u is four, we have e to the fourth,
minus when u is zero, we have e to the zero.
So the exact arc length would be the square root of five
divided by two, times the quantity e to the fourth,
minus e to the zero, which is equal to one.
So this'd be the exact arc length.
Let's also get a decimal approximation.
So we'd have the square root of five, right arrow,
divided by two, open parenthesis e,
second natural log brings up e raised to the power of four,
right arrow, minus one, close parenthesis, and enter.
So, the arc length is approximately 59.9246 units.
To go back to the graph one more time,
we just determined the arc length from this point,
when t equals zero
to this point, when t equals two.
Which again, will be the distance the point travels
from this point to this point,
as we see here along the curve.
I hope you found this helpful.