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PROFESSOR: Hi.

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Welcome back to recitation.

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We've been talking about
applications of integration,

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including finding the
areas between curves.

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So I have here a nice
little region that I like.

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I think it's kind of cute.

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So it's the region
between y equals

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sine x and y equals cosine x.

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And, you know, those two curves
cross each other over and over

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again.

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They wrap around each other.

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So I'm just interested
in the region

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between the two consecutive
points where they cross.

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So here, you know,
so at pi over 4

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we have sine x equals-- sine pi
over 4 equals cosine pi over 4.

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And at 5 pi over 4
they're also equal again,

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down in the third
quadrant there.

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So, the question is to compute
the area of this region

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that they bound between
those two points.

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So why don't you take a couple
of minutes, work through that,

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come back and we can
work through it together.

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All right, welcome back.

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So, from this picture
it's pretty easy

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to see what the region
of integration is.

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I mean what the
bounds on x will be.

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As we said, x has
to go the left,

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I mean, you could--
sorry, let me start over.

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You could do any two consecutive
intersection points you want,

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right?

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The area is the same in any
case because of the symmetry

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of these two functions.

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So, OK.

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But the first two that are
easiest for me to find are this

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pi over 4 and this 5*pi over 4.

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So I'm going to do those two.

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If you wanted, you
could have done it

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with a different pair
of consecutive points.

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But, once we've agreed that
sort of these first two

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are the ones I'm going
to do, then it's easy.

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I know that they're pi
over 4 and 5*pi over 4.

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So that's the
interval over which

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I'm going to be integrating.

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And then, what I
want to do is just

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view this region as cut into
a lot of little rectangles.

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And I want to integrate the
height of those rectangles

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in order to get the area
of the whole region.

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So in this case,
the upper curve is

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y equals sine x and the lower
curve is y equals cosine of x.

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So the height of a
little-- you know

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if I draw in a little
rectangle here--

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the height of that rectangle
is going to be sine

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x minus cosine x.

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Its width is dx.

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And then I add them
all up by integrating.

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So the area is just the integral
from pi over 4 to 5*pi over 4

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of sine x minus cosine x dx.

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Top minus bottom
to get the height.

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If you did it backwards;
if you wrote cosine

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minus sine, what you would
get is exactly the negative

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of the area.

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So it's, all right, from
pi over 4 to 5, pi over 4.

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So now, we just have
to integrate this.

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So integral of sine, the
function whose derivative

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is sine is minus cosine x.

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And the function whose
derivative is cosine is sine.

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So it's minus sine x between
pi over 4, 5*pi over 4.

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And OK, so now, we just
have to plug in the values.

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So this is equal to minus cosine
5*pi over 4 minus sine 5*pi

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over 4 minus minus cosine pi
over 4 minus sine pi over 4.

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And I'm sure you can work
out for yourself that this

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is equal to 2 times
the square root of 2

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if I haven't botched
anything terribly.

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So I'll end there.