Hi everyone. i need help integrating a function and im only in year 10 so i dunno where to search up how to do this. whilst im asking this ill post what im trying to do to check that ive been doing this right.

So the other day I was asked by my biology teacher to work out the surface area of a red blood cell. he gave me a "formula" of it which was just assuming it is a cylinder. I thought that was too rough so i tried to work out the exact area.

so i used $\displaystyle x^2+(y-2.5)^2=1^2$ to model the radius of the circle im trying to work out. im only trying to work out the inner bit of the torus

this turn into $\displaystyle y=-\sqrt{1-x^2}+2.5$, the - sign because im only trying to work out the inner bit of the torus. i will change to + when working out the outer bit.

times by $\displaystyle 2\pi$

to get the circumference $\displaystyle y=-2\pi\sqrt{1-x^2}+5\pi$

integrate to find surface area$\displaystyle \int{-2\pi\sqrt{1-x^2}+5\pi}dx$ in the limit 0.5 to 1 sorry i dont know how to put this in.

now im stuck as i dont know how to integrate $\displaystyle -2\pi\sqrt{1-x^2}$. i tried by part and substitution but my C3 and C4 textbook isnt helping me. is there like a chain rule for integration to integrate compound functions?

Thank in advance for any help! i hope that made sense.