find the aria between 2 curves

Question

Find the area of the region enclosed between 𝑦=2sin(𝑥)  and 𝑦=4cos(𝑥)from 𝑥=0to 𝑥=0.8pi. 

Hint: Notice that this region consists of two parts. 

Answer 1

Find the intersection 2sinx = 4cosx or tanx = 2 x = 1.107 rad, but, perhaps more useful cos(1.107) is 1/sqr(5) and sin(1.107) is 2/sqrt(5) (from right triangle with sides x=1, y = 2 and hyp = sqrt(5))

Split the interval of integration at this point because the cos function is > than the sine function below it and the reverse above it:

Area = integral from x= 0 to 1.107 of (4cosx-2sinx)dx + integral from 1.107 to .8π of (2sinx - 4 cosx)dx

A = (4sin(1.107) + 2cos(1.107) - 0 - 2) + (-2cos(.8π) -4sin(.8π) + 2cos(1.107) + 4sin(1.107))

I'll let you evaluate it. Please check as there are a lot of sign changes.