What is the area bounded by f(x) above the X axis and the area below the X axis?
X is in the domain from 0 to Pi
integral(Pi - x)*cos(x) = integral(Pi*cos(x) - x*cos(x))dx = Pi*integral(cos(x))dx - integral(x*cos(x))dx
integral(cos(x))dx = sin(x)
By integration in parts:
integral(x*cos(x))dx = x*sin(x) - integral(1*sin(x))dx = x*sin(x) - (-cos(x)) = x*sin(x) + cos(x)
Combining together:
Pi*integral(cos(x))dx - integral(x*cos(x))dx = Pi*sin(x) - x*sin(x) - cos(x)
f(x)=0 when x=Pi/2, x=Pi so f(x) crosses X axis at these 2 points
f(x)>=0 when x=[0, Pi/2]
f(x)<=0 when x=[Pi/2, P]
A = Area above axis X = Pi*sin(x) - x*sin(x) - cos(x) | from 0 to Pi/2
B = Area below axis X = -Pi*sin(x) + x*sin(x) + cos(x) | from Pi/2 to Pi
A = Pi*sin(Pi/2) - Pi*sin(0) - (Pi/2*sin(Pi/2) - 0*sin(0)) - (cos(Pi/2) - cos(0)) = Pi*1 - Pi*0 - (Pi/2*1 -Pi/2*0) - (0 - 1) = Pi - 0 - Pi/2 + 0 - 0 + 1 = Pi - Pi/2 + 1 = Pi/2 + 1
B = -(Pi*sin(Pi) - Pi*sin(Pi/2)) + (Pi*sin(Pi) - Pi/2*sin(Pi/2)) + (cos(Pi) - cos(Pi/2)) = -(Pi*0 - Pi*1) + (Pi*0 - Pi/2*1) + (-1 - 0) = -0 + Pi + 0 - Pi/2 - 1 - 0 = Pi - Pi/2 - 1 = Pi/2 - 1
Combined area under f(x) = A + B = Pi/2 + 1 + Pi/2 - 1 = Pi
Comments
integral(Pi - x)*cos(x) = integral(Pi*cos(x) - x*cos(x))dx = Pi*integral(cos(x))dx - integral(x*cos(x))dx
integral(cos(x))dx = sin(x)
By integration in parts:
integral(x*cos(x))dx = x*sin(x) - integral(1*sin(x))dx = x*sin(x) - (-cos(x)) = x*sin(x) + cos(x)
Combining together:
Pi*integral(cos(x))dx - integral(x*cos(x))dx = Pi*sin(x) - x*sin(x) - cos(x)
f(x)=0 when x=Pi/2, x=Pi so f(x) crosses X axis at these 2 points
f(x)>=0 when x=[0, Pi/2]
f(x)<=0 when x=[Pi/2, P]
A = Area above axis X = Pi*sin(x) - x*sin(x) - cos(x) | from 0 to Pi/2
B = Area below axis X = -Pi*sin(x) + x*sin(x) + cos(x) | from Pi/2 to Pi
A = Pi*sin(Pi/2) - Pi*sin(0) - (Pi/2*sin(Pi/2) - 0*sin(0)) - (cos(Pi/2) - cos(0)) = Pi*1 - Pi*0 - (Pi/2*1 -Pi/2*0) - (0 - 1) = Pi - 0 - Pi/2 + 0 - 0 + 1 = Pi - Pi/2 + 1 = Pi/2 + 1
B = -(Pi*sin(Pi) - Pi*sin(Pi/2)) + (Pi*sin(Pi) - Pi/2*sin(Pi/2)) + (cos(Pi) - cos(Pi/2)) = -(Pi*0 - Pi*1) + (Pi*0 - Pi/2*1) + (-1 - 0) = -0 + Pi + 0 - Pi/2 - 1 - 0 = Pi - Pi/2 - 1 = Pi/2 - 1
Combined area under f(x) = A + B = Pi/2 + 1 + Pi/2 - 1 = Pi