Final answer:
To find the area inside the curve r = 14 cos(θ) and outside r = 7 in the first quadrant, we integrate the functions ½(14 cos(θ))2 and ½(7)2 with respect to θ from 0 to π/2 and subtract the second from the first.
Step-by-step explanation:
To find the area of the region that lies inside of the curve r = 14 cos (θ) and outside of the curve r = 7, we use polar coordinates to set up an integral. The area inside a polar curve from θ = a to θ = b is given by the integral ∫ab ½ r2 dθ. Here, as we are looking for the area in the first quadrant (where cos(θ) ≥ 0), the limits of integration are from 0 to π/2. Calculating the integral will give us the area inside the first curve, and then we subtract the area inside the second curve.
To perform the calculation, we would find:
- The integral of ½ (14 cos θ)2 from 0 to π/2, and
- The integral of ½ (7)2 from 0 to π/2.
Subtracting the second integral from the first gives us the final area. Typically, one would use trigonometric identities and antiderivative formulas to compute the integrals.