This is a question designed to test your understanding of the underlying concepts as opposed to just being able to mechanically manipulate the functions.

If you have a value of the anti derivative F(x) at some point x=a and you want to find F(x) at x=b, you can find it with the equation F(b) = F(a) + <the area under f(x) between x=a and x=b>.

This problem gives you F(a) where a=0 and gives you f(x) between 0 and 9.

Some clarifications: f(x) = -2x/3 + 5 isn't technically correct. It happens to equal that when x is between 6 and 9, but the function is different outside of that range. Similarly, your equation for F(x) is only correct when x is between 6 and 9. The reason this matters is because F(0) = 2 doesn't mean C = 2. That only works if the function is the same all the way to x = 0, which it's not.

If you want to solve by integrating, you would have to integrate each section and find the right C for each section that makes the integrals all connect to each other.

Alternatively, you can use the property that F(b) - F(a) = the area under f(x) from a to b. I think that region from x = 4 to 6 is supposed to be a semicircle, so each section is a standard shape and you can calculate the area using geometry.

I recommend the math discord server. You might get faster responses there.

https://discord.gg/math