The area between two continuous functions #f(x)# and #g(x)# over #x in[a,b]# is:
#int_a^b|f(x)-g(x)|dx#
Therefore, we must find when #f(x)>g(x)#
Let the curves be the functions:
#f(x)=-4sin(x)#
#g(x)=sin(2x)#
#f(x)>g(x)#
#-4sin(x)>sin(2x)#
Knowing that #sin(2x)=2sin(x)cos(x)#
#-4sin(x)>2sin(x)cos(x)#
Divide by #2# which is positive:
#-2sin(x)>sin(x)cos(x)#
Divide by #sinx# without reversing the sign, since #sinx>0# for every #x in(0,π)#
#-2>cos(x)#
Which is impossible, since:
#-1<=cos(x)<=1#
So the initial statement cannot be true. Therefore, #f(x)<=g(x)# for every #x in[0,π]#
The integral is calculated:
#int_a^b|f(x)-g(x)|dx#
#int_0^π(g(x)-f(x))dx#
#int_0^π(sin(2x)-(-4sin(x)))dx#
#int_0^π(sin(2x)+4sin(x))dx#
#int_0^πsin(2x)dx+4int_0^πsin(x)#
#-1/2[cos(2x)]_0^π-4[cos(x)]_0^π#
#-1/2(cos2π-cos0)-4(cosπ-cos0)#
#1/2*(1-1)-4*(-1-1)#
#8#
