Calculating area under sine function in separate parts: f(x)=35sin(0.15(x-16))+30

[Riperonies]

Hey, so I'm trying to calculate the area under the function, f(x)=35sin(0.15(x-16))+30, from 16<x<36.207. But, I want to know the area above and below y=30 separately.
Obviously the area below 30 would just be (36.207-16=)20.207*30=606.21, as it's just a rectangle. Then you get the whole area under the function, and subtract 606.21 from that value to get the area above the rectangle, then you have both the areas, above and below 30 right?

So I did the integral of f(x), and got 233.333*cos(0.15(x-16))+30(x-16)+C. (I think that might be wrong? As I can't think of much else). Then use that to get f(36.207)-f(16)=839.2169 - 233.333=605.884. But how can the total area be less than just the area below y=30? That's where I'm stuck. Any thoughts on what's going on?

Here's the function you can paste into Desmos to have a look: 35\sin.15\left(x-16\right)+30\left\{15.486<x<36.207\right\}
Yes, it goes further left than x=16, but I'm still just working on 16<x<36.207 for this problem.

Cheers

 

[Dr.Peterson]

Hey, so I'm trying to calculate the area under the function, f(x)=35sin(0.15(x-16))+30, from 16<x<36.207. But, I want to know the area above and below y=30 separately.
Obviously the area below 30 would just be (36.207-16=)20.207*30=606.21, as it's just a rectangle. Then you get the whole area under the function, and subtract 606.21 from that value to get the area above the rectangle, then you have both the areas, above and below 30 right?

So I did the integral of f(x), and got 233.333*cos(0.15(x-16))+30(x-16)+C. (I think that might be wrong? As I can't think of much else). Then use that to get f(36.207)-f(16)=839.2169 - 233.333=605.884. But how can the total area be less than just the area below y=30? That's where I'm stuck. Any thoughts on what's going on?

The derivative of 233.333*cos(0.15(x-16))+30(x-16) is not 35sin(0.15(x-16))+30. (Try it and see.) Check your integration; if you don't see an error, show us your steps.

I also don't get the number you got when I evaluate f(36.207), assuming you took C=0. I assume you knew to use radians?

 

[Riperonies]

Solved!

Sorry! Should have canceled this thread already, just forgot. I ended up figuring it out, and yeah Dr. Peterson you were right, it was just a dumb little mistake in the integration hahaha. Cheers for your reply though!