Question

Find the area of the region that lies inside the first curve and outside the second curve. The first curve is given by the equation r = 10 cosθ and the second curve is given by the equation r = 5.

Answer 1

To find the area of the region that lies inside the first curve and outside the second curve, we need to determine the points of intersection of the curves and calculate the area between them using integration.

Step-by-step explanation:

To find the area of the region that lies inside the first curve and outside the second curve, we need to determine the points of intersection of the curves. Setting the two equations equal to each other, we have:

10 cosθ = 5

Dividing both sides by 10, we get:

cosθ = 0.5

Since cosθ = 0.5 at θ = π/3 and θ = 5π/3, these are the points of intersection. Now, we need to calculate the area between the curves. We can do this by integrating the equation of the first curve from θ = π/3 to θ = 5π/3 and subtracting the integral of the equation of the second curve from θ = π/3 to θ = 5π/3. The integral of r = 10 cosθ with respect to θ is:

∫(10 cosθ) dθ = 10 sinθ + C