9. let r be the shaded region enclosed by the graph of ( y=cos x ) and the x - axis for ( -\frac{pi}{2} leq…

9. let r be the shaded region enclosed by the graph of ( y=cos x ) and the x - axis for ( -\frac{pi}{2} leq x leq \frac{pi}{2} ), as shown in the figure. which of the following gives the volume of the solid generated when r is revolved about the horizontal line ( y = - 1 )? (a) ( pi int_{-\frac{pi}{2}}^{\frac{pi}{2}}((cos x + 1)-1)^{2} d x ) (b) ( pi int_{-\frac{pi}{2}}^{\frac{pi}{2}}left((cos x)^{2}-1\right) d x ) (c) ( pi int_{-\frac{pi}{2}}^{\frac{pi}{2}}left((cos x - 1)^{2}-1\right) d x ) (d) ( pi int_{-\frac{pi}{2}}^{\frac{pi}{2}}left((cos x + 1)^{2}-1\right) d x )

9. let r be the shaded region enclosed by the graph of ( y=cos x ) and the x - axis for ( -\frac{pi}{2} leq x leq \frac{pi}{2} ), as shown in the figure. which of the following gives the volume of the solid generated when r is revolved about the horizontal line ( y = - 1 )? (a) ( pi int_{-\frac{pi}{2}}^{\frac{pi}{2}}((cos x + 1)-1)^{2} d x ) (b) ( pi int_{-\frac{pi}{2}}^{\frac{pi}{2}}left((cos x)^{2}-1\right) d x ) (c) ( pi int_{-\frac{pi}{2}}^{\frac{pi}{2}}left((cos x - 1)^{2}-1\right) d x ) (d) ( pi int_{-\frac{pi}{2}}^{\frac{pi}{2}}left((cos x + 1)^{2}-1\right) d x )

Answer

Explanation:

Step1: Recall the Washer - Method formula

The formula for the volume (V) of a solid of revolution using the Washer - Method when rotating about the line (y = k) is (V=\pi\int_{a}^{b}\left[(R(x))^{2}-(r(x))^{2}\right]dx), where (R(x)) is the outer radius and (r(x)) is the inner radius.

Step2: Determine the outer and inner radii

When rotating the region (R) (bounded by (y = \cos x) and (y = 0) for (-\frac{\pi}{2}\leq x\leq\frac{\pi}{2})) about the line (y=-1):

  • The outer radius (R(x)) is the distance from the line (y = - 1) to the curve (y=\cos x). Using the distance formula (d=y_2 - y_1), we have (R(x)=\cos x-(-1)=\cos x + 1).
  • The inner radius (r(x)) is the distance from the line (y=-1) to the (x) - axis ((y = 0)). So (r(x)=0-(-1)=1).

Step3: Set up the integral

Substitute (R(x)) and (r(x)) into the Washer - Method formula (V=\pi\int_{a}^{b}\left[(R(x))^{2}-(r(x))^{2}\right]dx). Here (a =-\frac{\pi}{2}), (b=\frac{\pi}{2}), (R(x)=\cos x + 1), and (r(x) = 1). The integral becomes (V=\pi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left[(\cos x + 1)^{2}-1^{2}\right]dx=\pi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left[(\cos x + 1)^{2}-1\right]dx)

Answer:

D. (\pi\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left[(\cos x + 1)^{2}-1\right]dx)