Today we're going to talk about four important applications of integration.
After all, what good is knowing how to solve integrals, if you don't know what you can
do with them?!
In a typical calculus course, you'll usually study how integrals can be applied to Physics,
or Geometry, or Biology.
Since those are already real-world applications, it's easier to see how integrals can be used
in those fields.
But what about some of the more abstract applications, like average value, area between curves, arc
length and surface area of revolution?
Those are the applications I want to talk about today.
By the end of this video you should have a better understanding about how these concepts
can be applied in the real world.
AVERAGE VALUE
Let's start with average value.
Remember when you first learned about finding an average, or a mean?
What did you have to do to find the mean of 5 numbers?
Well, you'd add them all together and then divide by 5, because that's how many numbers
you started with.
Now think about a simple graph, like this, defined on the interval x=0 to x=10 (show
straight, horizontal line).
The value of this function everywhere on the interval is 7, because no matter where I test
inside the interval, the height of the function, or the distance between the function and the
x-axis, is 7.
When we're trying to find average value, we're basically just looking for the average height,
so we can do this the same way we took the average of the 5 numbers.
We test the function at 5 x-values, get the value, or height of the function there, then
add the heights together and divide by 5.
This tells us that the average value of the function on the interval is 7.
But what if we have a more complicated function, like this (show sin(x)).
How do we find the average value of this function on the interval x=0 to x=4pi?
We could do the same thing, and test randomly at 5 values of x to try to estimate the average,
but we want to be more accurate than that.
I won't take you through the proof today, but if we write a formula to test the function
at not 5, not 500, not 5000, but infinite values of x, we get this formula for the average
value of a function on the interval a to b.
If we use this formula, we'll get back a constant value, which we'll reveal for this function
would be 0.
If we draw the line y=0 over the top of our original function, we see that it does in
fact appear to be the average of the function over the interval.
So how can we apply this to the real world?
Well, given a function for temperature in terms of time, we could find the average temperature
inside an oven during a given time period.
Or, given a function for growth over time, we could find the average yearly growth of
a tree over a given time period.
Or, given a function for water volume in terms of time, we could find the average volume
of water that falls over a waterfall in a given time period.
AREA BETWEEN CURVES
What about area between curves?
This is another very common application of calculus.
Using this formula we can find the area enclosed by two graphs like this.
If we call our higher function (literally the graph that's above the other one) f(x),
and the lower function (the one underneath) g(x), then our integral formula starts to
make sense.
Remember that the properties of integrals tell us that we can split up this integral
into two separate integrals, like this.
Then we can see that we're finding the area under f(x) all the way to the x-axis, and
subtracting from that the area under g(x) all the way to the x-axis.
When we do that, we're just left with the area between the curves, this portion right
here.
Which means, in order to solve an area between curves problem, all you need to do is find
the points of intersection of your two curves, which will become the limits of integration,
identify which function is higher and which one is lower, and then plug all the pieces
into the formula.
Remember, in order to figure out which function is higher and which is lower, you just need
to pick an x-value between the points of intersection and plug that value into both functions to
see which value is larger than the other (the larger value is the higher function).
So how can we use this in the real world?
Well, let's say you're a farmer, and you need to find the area of your field in order to
know how much fertilizer to buy.
That's no problem if your field is square like this, because in that case you'd just
take length times width to get the area.
But what if your field is irregularly shaped, like this?
If you could find a function to model both sides of the perimeter, you could use area
between curves to find the area of the field.
ARC LENGTH
Here's one that's a little more complicated: arc length.
How do we find the length of the curve sketched out from a function?
Well, if our function is a straight line, and we want to find the length on the interval
x=0 to x=10, it's easy, the length is 10.
But how do we find the length of this line if the line looks like this?
We can't just straighten it out and find the length like we did with the line.
We could draw straight line segments that approximate the curve, and then add up the
length of each line segment.
That would be easy, but time tedious, time consuming, and unfortunately not very accurate.
The more line segments we use, the more accurate our estimate will be.
Therefore, if we want an accurate value for arc length, we need to use an infinite number
of line segments to measure the length.
When we do that, our formula for arc length on the interval a to b looks like this, and
we can use it to find the length of the curve on an interval, the same as if we had taken
the curve and stretched it out until it was a straight line and then measured it against
the x-axis.
So, you know what's coming...
how do we apply this to the real world?
Imagine you're an engineer and you build bridges.
You're newest project is a bridge over a big river.
You want to build a tower on each side of the river and hold up the bridge with cables
that connect the towers to the roadbed.
Using the arc length formula, you can figure out how long each of these cables need to
be.
Knowing the length means you can build the cable to the right length before you install
it on the bridge.
SURFACE AREA OF REVOLUTION
We've got just one more application to talk about, and that's surface area of revolution.
We're already familiar with finding the surface area of normal geometric shapes, like a cube
or a sphere.
Surface area of revolution is generated by taking a function and revolving it around
an axis to create a three-dimensional surface from a two-dimensional graph.
Once we have a three-dimensional figure, how do we find the surface area of that figure?
To answer this question, let's go back to arc length.
To approximate arc length, we said we could use a finite number of straight line segments.
If we take those line segments and rotate them around an axis, we get a surface area
approximation of the three-dimensional figure.
Each of these segments is called a frustum.
To find the surface area of one frustum, we use the formula A=2(pi)rl, where r is the
radius of the frustum and l is the length of the frustum.
This should make sense, since we know that 2(pi)r gives the circumference of a circle,
so multiplying that value by the length of the frustum would give us the surface area.
Since l is the length of the frustum, and nothing more than the arc length of the function
that we rotated around the axis, we can replace l in this formula with the arc length formula
we found before.
If we make the length of the frustums shorter and shorter, so that there are more and more
frustums approximating the surface area until eventually the number of frustums goes to
infinity, we can prove that the radius can be represented by the function itself, and
we get an integral formula for surface area.
Keep in mind that our surface area formula changes depending on the axis of rotation
and whether our function is defined for y in terms of x, or for x in terms of y.
You can use this table to help you remember which formula to use.
So how do we use this in the real world?
Imagine you're an artist who makes lamps.
Some of the bases you design are very elaborate.
You like to experiment with different shapes, but you always need to consider the cost of
the paint required to coat the base when you're finished making the shape.
If you turn the base on its side and can find a function that defines the curve of the base,
you can rotate the curve around the axis of the base and then use the surface area of
revolution formula to find the surface area of your three-dimensional lamp base.
Think also about the coating on a baseball bat, the finish on a table leg, or the fabric
covering a cylindrical pillow.
Finding the cost of all of these would require you to apply the surface area of revolution
formula.
Keep in the back of your mind that all of these applications will come back around when
we get to multivariable calculus, but instead of starting with functions in two variables
that are graphed on a two-dimensional coordinate system, we'll start with a function in three
variables and graph it in a three-dimensional coordinate system, but we'll save that for
later.