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.
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1 Answer
May 30, 2018

#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
James