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?

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.

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Answer 1 · 2018-05-30T04:03:14

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

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
James