WEBVTT
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We want to evaluate the value of this integral,
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and we want to interpret it as an area.
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So first, let's draw a quick sketch. Let's
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let's take a look. We're going from X equals
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zero toe X equals pirate, too. First,
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let's sketch sine X. So sign it achieves its
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minimum a pirate too. And coast of two X
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. That's just a compressed division of cosa vex eso
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when access pirate to that becomes close if pie which
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is minus one. So that's what coast of two
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X looks like. And we're only going from zero
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to private, too. Now, since we're integrating
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the absolute value of this, uh, the absolute
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value is just concerned with what is the difference between
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these two functions? The actual positive distance. So
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when we integrate, what we're going to be calculating
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is the actual area enclosed bay. Ah, by
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the difference, the area of these two regions where
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both of these regions are considered to have positive area
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. So that's where the interpretation of this integral we're
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going to break up the integral at this point because
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to the left of this point, ah, coast
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of two X is greater than sign of X.
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So we're going to take the negative value from the
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absolutely integral and then to the right sign. Xs
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positive. So to the right, we can just
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remove the absolute values essentially as if it wasn't wasn't
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there. So we're gonna have to figure out what
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this X value is. So to do that,
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we're going to find the intersection of sine X and
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coast of two X. Let's do that Sine X
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equals coasts of two x Here we're going to use
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a triggered energy coast of two X is equal to
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one minus sine squared x So let's label that trig
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identity Bring everything to one side sine squared acts Oh
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, there's a to here one two signs squared X
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plus sign X minus one. This is actually a
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quadratic in sine X So if we think of Sine
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X is a than sine squared X is a squared
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. So what we have here is zero equals to
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a squared, plus a minus one so we can
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solve this. Using the quadratic formula which gives us
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A is equal to what we get to values.
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The 1st 1 is 1/2 and the 2nd 1 is
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minus two. Um, so at this point,
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over here is going to be at 1/2 because clearly
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from the graph, um, this value is positive
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. Eso this is a equals 1/2 which tells us
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that sine X equals 1/2. So how do we
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figure out the value? We remember our special triangle
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, which has, um, side length of one
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to and square three. This angle over here is
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pi over three. This angle is pi over six
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. So this triangle is really going to help us
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in this in this question. So we see that
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a sign of which angle is 1/2 That's going to
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be pirates six. So x is equal to pi
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over six. So we can label that over here
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. X equals pi over six. And then now
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we can break up this interval at private six.
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So is going to be integral from zero to pi
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over six of our top function minus their bottom function
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. Here, the top function is coast of two
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x coast of two x, and the bottom function
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is sine x d x and then to the right
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of pi over six. So pi over six pi
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over six to pi over, too. Ah,
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we see that sign access greater So sign X minus
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coast of two x. So that's what the absolute
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value tells us. It says that when the value
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inside the absolute value is negative, you have to
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take the opposite sign. So that's what we did
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in the first piece here. Okay, So to
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figure out this total area, we need toe evaluate
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this integral these two intervals. So let's add a
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new page and write it out. Area is equal
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to first. We're going from zero to pi over
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six of coast of two x minus sign X dx
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and then we're doing the second inter go from pi
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over six. The pie over too of sine X
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minus coast of two x dx. So let's just
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Valerie these into girls. The anti derivative of coast
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of two eggs is half sign of two X.
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If it's not clear to you how we can get
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that, uh, instantly, you could do a
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U substitution for U equals two X equals two X
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Uh, okay. And in the anti derivative of
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sine X of negative Sign X is co sex.
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We're going from zero to pi over six. Uh
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, here, the intruder votive of sign is negative
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coasts and then the anti drew of negative coast of
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two x is negative. Half sign of two x
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, So it's the same story as in the first
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, Integral. If it's not clear to you,
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you could do a U substitution eso here we're going
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from pi over six to pi over too. Okay
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, let's plug in the numbers, son of two
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times Private six. So that sign of pie or
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three, we take a look at our triangle over
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here. Sign of pie or three is Route three
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over two, and then we have this half.
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So half read. Three over two. Ah,
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plus Coast of Piper six. Again, let's check
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this triangle. Coast of Perverse six is Route three
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over two minus and we plug in zero. So
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sign of zero is zero coast of zero is one
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. So zero plus one. Okay, Plus,
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now we plug in pi over too. Ah,
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coast of poverty, too, is zero. And
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then sign of pi over two times to that sign
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pie. So that's just zero. And now we
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plug in pi over six coast of private. Six
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. Let's check the strangle again. Coast of Private
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six is Route three over two. So we have
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this negative. We're three over two minus and 1/2
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sign of two times power Six. So that sign
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of power three, we checked the triangle. Sign
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of pi Over three is retriever too. We're three
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over two. Okay, well, we have these
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zeros aren't there. And then after we simplifying,
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collect everything, this works out to be three times
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screwed. Three over two, minus one. So
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that's the area of the enclosed region.