The polar curves r=1-cos theta and r=1+cos theta are shown in the figure above. Which of the following expressions gives the total area of the shaded region
Let's denote the area enclosed by the two curves as A. To find the total area of the shaded region, we need to find the area enclosed by the curves and then subtract the overlapping area.
The points of intersection of the two curves occur when:
1 - cos(theta) = 1 + cos(theta)
2cos(theta) = 0
theta = pi/2, 3pi/2
Now, we can calculate the area enclosed by the curves using the formula for polar area:
A = ∫[θ1,θ2] (1+cos(theta))^2 - (1-cos(theta))^2 d(theta)
A = ∫[pi/2,3pi/2] (2cos(theta) + 2) d(theta)
A = 2 ∫[pi/2,3pi/2] (cos(theta) + 1) d(theta)
A = 2[sin(theta) + theta] [pi/2,3pi/2]
A = 2[(sin(3pi/2) + 3pi/2) - (sin(pi/2) + pi/2)]
A = 2[(-1 + 3pi/2) - (1 + pi/2)]
A = 2[3pi - 4]
A = 6pi - 8
Therefore, the total area of the shaded region is 6pi - 8.