So in this problem we are given an equation that demonstrates the monthly sales of snow blowers. And this question in part is asking how many snowblowers are sold during an entire year. So we need to add up all of the snow blowers sold in january and february and March and april all the way through december. So the best way to do that would be to find the anti derivative aka the area under the curve, aka the sun. So we're going to go ahead and evaluate this equation. I split it into two anti derivatives because there's a plus sign between the 15 and then the trick function. And then we're going to evaluate this anti derivative using the first fundamental theorem of calculus between january and december. So let's go ahead and get started. The anti derivative of 15 Is just 15 groups. They don't actually need to write this symbol anymore is 15 T. And like I said, that's going to be evaluated between zero and 12, which they are telling us in the problem. Plus now when it comes to this trade function, um this is sign of this whole thing in here. And so if we were to take the derivative of this, it would require chain rules. So now when we're taking the anti derivative we need to use U. Substitution because we know what the anti derivative of sine of X. Is. But it gets a lot more complicated when you have signs of two X. Or in this case sine of pi over six times the quantity X minus eight. And so in order to deal with that more complicated um step we go ahead and use what we call U substitution. So I'm gonna come over here and and make that long thing equal to you So I can come down here and rewrite this. This is now six sign of you D. T. So I've taken care of the fact that now I know what the anti derivative of sine of you is, but I'm still multiplying by D. T. And so my variables at the moment don't match. So I need to rearrange some things over on the right here to get D. T. In terms of the you. Instead I need my variables to match. So I'm gonna go ahead and take the derivative of you, which is do you over D. T. And then this is really Hi over six T minus eight, pi over six. So the derivative of pi over 60 is just pi over six. And then the during a pi over six, that's a constant term. So the derivative of that is zero. Like I said, I want to replace the DT with A D. U. So I need a equation that starts with DT equals. so I can do a direct substitution rearranging this equation up here. I can cross multiply and I get DT is equal 2 6 over pie. Do you? So on this line I'm gonna be rewriting this 15 TA couple more times D. T. Is the same thing as six over pi D. You. So I can do a direct replacement here. And now I have my trig in terms of something I know how to take the anti derivative of and I have matching variables. So I can go ahead and do the anti derivative. Now I'm going to bring the coefficient out in front until six times six over pie is 36 over pie. The anti derivative of sine is negative co sign. So I'm going to make that a negative And again this is being evaluated between zero and 12. Now that zero and the 12 is in terms of T. It is not in terms of you. So, we need to plug this value back in now. Everything is matching as it should be. Let's go ahead and finish the algebra. I'm going to use the first fundamental theorem of calculus, which means I plug in the larger number in this case 12 and then subtract off the smaller number. So we can do 15 times 12 minus 15 time zero minus. Oops. I'm gonna write that 36 over pie outside. All right. So, let's do some mental math here plugging in 12 for T. So 12 minus eight is four, that is four pi over six. Which is the same thing as two Pi over three. And then I can go ahead and plug it in again, plugging in 00 -8 is negative eight over six. So that is co sign of negative 8/6 which is the same thing as for over three. Okay 15 times 12. 180 plus 36 over a pie. And then coastline of 2/3 Using the unit circle is negative 1/2 Minour coastline of negative 4/3 Is also negative 1/2. So these two end up cancelling each other out. Now you have one half plus one half becomes zero. You are multiplied by anything is just zero. So the answer Is 180. So that was the answer for part a asking for the total monthly sales in an entire year. Part B is doing something similar but just july through december. So we can scroll down here and instead of redoing the whole thing we can take this line And instead of evaluating from 0 to 12 we can evaluate it from 7 to 12. So let me rewrite this 15 T evaluated from 7 to 12 minus 36 pie over five co sign of Pi over six. Two minus eight Evaluated from 7 to 12. So using the first fundamental theorem of calculus again we're going to do 15 times 12 -15 times seven. I put that back, it's just too late. Make it super clear minus. Okay, so as I'm writing this, I'm actually realizing that I had one too many minus signs up here so I can go ahead and take one of these out. My apologies and then this should be a positive a negative but it didn't end up mattering because things ended up canceling out here to zero. So it did not affect the answer but I'm going to get it right this time in the green so minus 36 over pie. And then inside the brackets we've got co sign of. Okay, so 12 minus eight is 44 pi over six is two pi over three minus both. Sign Of 7 -8 is a -1. So negative pi over six. Mhm. This was 1 80 15 times 12 was 1 80 15 times seven Is 105. So we get the difference here to be 75 -36 over pie. Co sign of two pi was To buy or three was negative 1/2 and co sign of negative pi over six is Radical 3/2. Yeah. All right. So I'm gonna go to my calculator to solve the rest of this and I plugged all of this in and I got 90 0.65 And so let's just double check and make sure that makes sense. In the green we are only talking about july through december, whereas in the purple, we were talking about the entire year. So july through december should be less than the purple. And it is, it actually looks like it's cut right in half, which is awesome. So if someone is talking about monthly sales or something like that in context, it sounds like they're being pretty consistent in terms of one half of the year and the other. You can also double check this using your calculator. I actually walk through that in problem. 56, much more in depth problem. 56 uses a graphing calculator and does not do the algebra. So check that problem out. If you would like to see how to double check this on your calculator.