Find the volume of the solid that results when the regionenclosed by the given curves is revolved about the x-axis. ■11. y = √25 − x2, y = 3

To find the volume of the solid that results when the region enclosed by the given curves is revolved about the x-axis, we can use the method of cylindrical shells.

First, let's sketch the region enclosed by the curves. The curve y = √(25 - x^2) represents the upper boundary of the region, while the line y = 3 represents the lower boundary. The region is bounded by x = -5 and x = 5.

Next, we need to determine the limits of integration. Since we are revolving the region about the x-axis, the limits of integration will be the x-values where the curves intersect. To find these points, we set the two equations equal to each other:

√(25 - x^2) = 3

Squaring both sides, we get:

25 - x^2 = 9

Rearranging the equation, we have:

x^2 = 16

Taking the square root of both sides, we get:

x = ±4

So the limits of integration are x = -4 and x = 4.

Now, let's set up the integral to find the volume. The volume of a cylindrical shell is given by the formula:

V = ∫[a,b] 2πx(f(x) - g(x)) dx

where [a,b] represents the limits of integration, f(x) is the upper curve, g(x) is the lower curve, and x represents the variable of integration.

In this case, the upper curve is y = √(25 - x^2) and the lower curve is y = 3. Plugging these values into the formula, we have:

V = ∫[-4,4] 2πx(√(25 - x^2) - 3) dx

Now, we can integrate this expression to find the volume.