Find the volume of the solid that results when the regionenclosed by y = √x, y = 0, and x = 9 is revolved aboutthe line x = 9

To find the volume of the solid that results when the region enclosed by y = √x, y = 0, and x = 9 is revolved about the line x = 9, we can use the method of cylindrical shells.

Step 1: First, let's sketch the region enclosed by the given curves. The region is a triangle with vertices at (0,0), (9,0), and (9,3).

Step 2: Next, we need to determine the height of the cylindrical shells. Since we are revolving the region about the line x = 9, the height of each shell will be the difference between the x-coordinate of the point on the curve y = √x and the line x = 9. In this case, the height of each shell is 9 - x.

Step 3: Now, we need to find the radius of each cylindrical shell. The radius is simply the y-coordinate of the point on the curve y = √x. In this case, the radius is √x.

Step 4: The volume of each cylindrical shell can be calculated using the formula V = 2πrh, where r is the radius and h is the height. In this case, the volume of each shell is 2π(√x)(9 - x).

Step 5: To find the total volume of the solid, we need to integrate the volume of each shell over the interval [0,9]. The integral is given by ∫[0,9] 2π(√x)(9 - x) dx.

Step 6: Evaluating the integral, we get the total volume of the solid as V = ∫[0,9] 2π(√x)(9 - x) dx.

Step 7: Finally, we can calculate the value of the integral to find the volume of the solid.

Note: The steps provided above assume basic knowledge of calculus and integration.