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Hovedindhold

Aktuel tid:0:00Samlet varighed:8:27

For fun, let's try to
approximate the area under the curve y is
equal to the square root, the principal root, of x minus
1, between x is equal to 1 and x is equal to 6. So I want to find
this entire area. Or I want to at least
approximate this entire area. And the way I'll do
it is by setting up five trapezoids of equal width. So this will be
the left boundary of the first trapezoid. This will be its right
boundary, which will also be the left boundary of
the second trapezoid. This will be the right boundary
of the second trapezoid. This is the right boundary
of the third trapezoid. This'll be the right boundary
of the fourth trapezoid. And then finally, this
will be the right boundary of the fifth trapezoid. And since we're going from 1 to
6, so we're traveling 6 minus 1 in the x direction, and I want
to split it into five sections, the width of each trapezoid is
just going to be equal to 1. And so if we say that the width
of a trapezoid is delta x, we just can now say the
delta x is equal to 1. So let's set up our trapezoids. So the first trapezoid is
going to look like that. It's going to go like that. Actually, it's going to
be a triangle, not really a trapezoid. Then the second trapezoid
is going to look like this. I guess you could
say a trapezoid where one of the sides has length
0 turns into a triangle. And then the third trapezoid
is going to look like this. And then the fourth trapezoid
is going to look like that. And then finally, you
have the fifth trapezoid. So let's calculate the
area of each of these, and then we will have our
approximation for the area under the curve. So let's do trapezoid--
or I really should say triangle-- this first shape,
whatever you want to call it. What is the area of
that going to be? Well, the area of a
trapezoid-- and you'll see this will just turn into
the area of a triangle-- it's the average of the
heights of the two sides of the trapezoid, the
way we've looked at it-- or you could say the average
of the heights of the two parallel sides, I guess
is the best way to say it. So f of 1-- that's the
height here-- plus f of 2, all of that over
2, and then we're going to multiply it
times our delta x. Actually, let me do that
in that same red color to show you that this is the
area of that first trapezoid, so times delta x. And as you see right over here,
if you look at it, the f of 1 is just going to be 0. So you're going to
have f of 2 times-- so it's going to be this height
times this base times 1/2, which is just the
area of a triangle. Let's look at the
second trapezoid, trapezoid two right over here. What is its area going to be? Well, it's going to
be f of 2 plus f of 3. f of 2 is this height. f of 3 is this height. So we're taking the average
of those two heights-- divided by 2; that's the
average of those two heights-- times the
base, times delta x. And then let's do
trapezoid three. I think you're
getting the idea here. Trapezoid three is
going to be f of 3 plus f of 4 divided
by 2 times delta x. And then-- let's see. I'm running out of colors. This is trapezoid
four right over here. So plus f of 4 plus f of 5, all
of that over 2, times delta x. And then we have
our last trapezoid, which I will do in yellow. So this is trapezoid
number five. I'll scroll down a little bit,
get some more real estate. So it's going to be plus-- I'll
just write the plus over here-- plus f of 5 plus f of 6
over 2 times our delta x. So let's see how we can
simplify this a little bit. All of these terms,
we have a 1/2 delta x, so let's actually
factor that out. So remember, this is our
approximation of our area. So area is
approximately-- remember, this is just a
rough approximation. It's very clear--
actually, you might say, this is pretty good
using the trapezoids, but it is pretty
clear that we are letting go of some of the area. We're letting go of that area. We're letting go of some
of this right over here. You can barely see it. Some of this right over
here, you can barely see it. But we are-- this is going
to be, it looks like, an underestimate, but it
is a decent approximation. But let's see if we can
simplify this expression. So it's approximately
going to be equal to, I'm going to factor
out a delta x over 2. And then what I'm
left with-- and I will switch to a neutral
color-- if I factor out a delta x over 2, then
I have just an f of 1. And then I have two f of
2's, so plus 2 times f of 2. And I'm doing this because
you might see a formula that looks something like this
in your calculus book, and it's not some
mysterious thing. They just really summed up
the areas of the trapezoids. And then we're going to have
two f of 3's-- plus 2 times f of 3's. Plus we're going to have two f
of 4's-- plus 2 times f of 4. And then we're going to have two
f of 5's-- plus 2 times f of 5. And then finally, we're going to
have one f of 6-- plus f of 6. If you were to
generalize it, you have one of the first
endpoint, the function evaluated at the first
endpoint, one of it at the very last endpoint, and then two
of all of the rest of them. But this is just the
area of trapezoids. I'm not actually a big fan
of when textbooks write this. Because when you see this,
it's hard to visualize the trapezoids. When you see this,
it's much clearer how you might visualize that. But with that out of the way,
let's actually evaluate this. Lucky for us, the
math is simple. Our delta x is just 1. And then we just have to
evaluate all of this business. f of 1, let's just
remind ourselves what our original function was. Our original function was
the square root of x minus 1. So f of 1 is the square
root of 1 minus 1, so that is just going to be 0. This expression
right over here is going to be 2 times the
square root of 2 minus 1. The square root of
2 minus 1 is just 1, so this is just going to be 2. Actually, let me do it
in that same-- well, I'm now using the purple
for a different purpose than just the first trapezoid. Hopefully, you realized that. I was just sticking
with that pen color. Then f of 3. 3 minus 1 is 2--
square root of 2. So the function evaluated at
3 is the square root of 2. So this is going to be 2
times the square root of 2. Then the function
evaluated at 4. When you evaluate it at 4,
you get the square root of 3. So this is going to be 2
times the square root of 3. And then you get 2 times the
square root of 4-- 5 minus 1 is 4. 2 times the square
root of 4 is just four. And then finally, you get f of
6 is square root of 6 minus 1, is the square root of 5. And I think we're now
ready to evaluate. So let me get my handy TI-85
out and calculate this. So it's going to
be-- well I'm just going to calculate--
well, I'll just multiply. So 0.5 times open
parentheses-- well, it's a 0. I'll just write it, just
so you know what I'm doing. 0 plus 2 plus-- whoops. Lost my calculator. Plus 2 times the square root of
2 plus 2 times the square root of 3 plus 4-- I'm almost done--
plus the square root of 5-- so let me write
that-- gives me-- now we are ready for our
drum roll-- it gives me-- and I'll just round it-- 7.26. So the area is
approximately equal to 7.26 under the curve y is equal
to the square root of x minus 1 between x
equals 1 and x equals 6. And we did this
using trapezoids.

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