Find the points where the tangent line is horizontal or vertical. Find the area under one period of the curve.

x= r(deta - cos deta)
Y= r(1- sin deta)

See:

http://www.jiskha.com/display.cgi?id=1283050369

To find the points where the tangent line is horizontal, we need to find the values of θ (deta) where the derivative of the curve with respect to θ is equal to zero. When the derivative is zero, the tangent line is horizontal.

Let's find the derivative of x and y with respect to θ:

x = r(θ - cos(θ))
y = r(1 - sin(θ))

To find dx/dθ, we apply the chain rule:

dx/dθ = r(1 + sin(θ))

To find dy/dθ, we also apply the chain rule:

dy/dθ = -r*cos(θ)

Now, set dx/dθ equal to zero and solve for θ:

0 = r(1 + sin(θ))

Since r is always positive, we can divide both sides of the equation by r:

0 = 1 + sin(θ)

Solving this equation for θ will give us the values of θ at which the tangent line is horizontal.

Similarly, to find the points where the tangent line is vertical, we need to find the values of θ where the derivative of y with respect to θ is equal to zero:

0 = -r*cos(θ)

Again, dividing by -r:

0 = cos(θ)

Solving this equation will give us the values of θ at which the tangent line is vertical.

To find the area under one period of the curve, we need to find the integral of y with respect to θ over one complete period, and then take the absolute value of the integral since the curve can go above and below the x-axis.

The period of the curve is determined by the range of θ over which the curve repeats. Since the equations for x and y involve trigonometric functions, we need to determine the range of θ that generates one complete period.

For x = r(θ - cos(θ)), we can find the range by setting θ = 2π and solving for θ:

2π = θ - cos(θ)

Solving this equation for θ will give us the upper and lower bounds of the period.

Similarly, for y = r(1 - sin(θ)), we can find the range by setting θ = 2π and solving for θ:

2π = 1 - sin(θ)

Solving this equation for θ will give us the upper and lower bounds of the period for y.

Once we have the upper and lower bounds of the period, we can integrate y with respect to θ over that range to find the area under the curve.