Question

Find the area bounded by `f(x)=sinx` and `g(x)=cosx` from `x=pi/4` to `x=((5pi)/4.`. Make an accurate sketch of the graphs on the axis below?

Find the area bounded by `f(x)=sinx` and `g(x)=cosx` from `x=pi/4` to `x=((5pi)/4)`. Make an accurate sketch of the graphs on the axis below.

Answer 1

`color(blue)[A=int_(pi/4)^(5pi/4)sinx-cosx*dx=2sqrt2]`

Explanation:

The Area due to `"x-axis"` between two curves given by:

`color(red)[A=int_a^by_2-y_1*dx`

The interval of our integral `x in [pi/4,5pi/4]`

now let set up the integral:

`A=int_(pi/4)^(5pi/4)sinx-cosx*dx=[-cosx-sinx]_(pi/4)^(5pi/4)`

`[(-sin(5pi/4)-cos(5pi/4))-(-sin(pi/4)-cos(pi/4))]`

`[2/sqrt2+2/sqrt2]=4/sqrt2=2sqrt2`

show the wanted area below(shaded):
`y=sinx` blue curve
`y=cosx` green curve