 Hi everyone, I have an example here for you of another problem in finding the area between two curves. This one's a little different because number one it's a trig function but it makes use of some of those characteristics of the sine and cosine curves in that they intersect obviously infinitely many times. And what we want to do is find the area of one of those regions. So let's take a look at it on our graphing calculator. To make sure you're in radiant mode when you go to do your graph it would be best to graph it with the zoom 7 zoom trig. So the first one you are seeing here is the sine curve and the red one coming up as your cosine curve. So if you take a look at those regions it kind of looks like a DNA molecule of sorts. We want one of those regions. You really could pick any one of those that you choose. You will have to find the points of intersection or just think of your unit circle values and where those sine and cosine curves would intersect. So really any one of these regions could work and you would just set up your integral accordingly. So I'm going to pick the one right here in the middle sort of right here kind of in this little corner of the second quadrant down into the third quadrant and I got a little little part there in the first quadrant. So I'm going to have to find these two points of intersection one in the third quadrant one over here in the first quadrant. And again there's a couple different ways you could do that. You could use your graphing calculator and get decimal approximations or you can think of your unit circle. So I think we should do that because it's a good exercise in the unit circle. So let's find the easy one up here in the first quadrant. So think about where sine and cosine are the same value. That happens at pi over 4 and since this would be a dx problem think again of the orientation of your rectangles, your representative rectangle. It would go vertically so therefore it's a dx problem. Therefore our limits of integration have to be x values. So the one limit of integration is going to be pi over 4. Then we have to think of the one over here in the third quadrant. So down in the third quadrant think about where it is that sine and cosine are the same. Once again your reference angle is going to be pi over 4. You do have to think in terms of negative angles though so it would be negative 3 pi over 4. Now of course you could go ahead and do decimal values for those. Find the points of intersection. You would have to store those in your calculator though to use. So sometimes it is actually easier to use the exact values. So take a look at the way we're going to have to set this up because remember our representative rectangle on the upper end it's going to be hitting this red curve. Remember the red curve is your cosine and the one at the bottom it's hitting the blue curve is your sine. So that's the way in which we're going to have to set up our integral. So let's go ahead and do that. So you could go ahead and do your antiderivative by hand if you'd like. So your antiderivative cosine of course remember is sine. Antiderivative sine is going to be negative cosine so that's going to be minus a negative cosine so it turns into plus. This would be a great example of one you might have to do by hand but or we could use our graphing calculator. So let's go ahead and try that. So my lower limit was negative 3 pi over 4. My upper limit was pi over 4 and cosine of x minus sine of x dx. So it looks like that. So let's see what we get. 2.828 that's actually 2 square root of 2 if you were to have done the exact value. The exact value of course is what you would have gotten if you did this by hand and evaluated your antiderivative that way.