Problem 35 Hard Difficulty

Evaluate the integral and interpret it as the area of a region. Sketch the region.

$ \displaystyle \int_{0}^{\frac{\pi}{2}} \mid \sin x - \cos 2x \mid dx $

Answer

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Video Transcript

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