Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 9.
y = x, y = 0, y = 8, x = 9
see the first problem done for "Mark" a few posts below.
Hey Im quite stuck on this problem and I can't find the post below.
.Which of the following represents the volume of the solid formed by revolving the region bounded by the graphs of y =x^3, y = 1, and x = 2, about the line x = 2?
To find the volume of the solid generated by revolving the region bounded by the given equations about the line x = 9, we can use the method of cylindrical shells. Here's how you can do it step by step:
Step 1: Sketch the region bounded by the given equations. The region is a triangle with a base on the x-axis, bounded by y = 0 and y = 8, and with its vertex at the point (9, 9).
Step 2: Identify the limits of integration for x. Since the triangle is bounded by x = 9 on the right, we can integrate from x = 0 to x = 9.
Step 3: For a given value of x, find the height of the triangle at that x-value. Since the triangle is defined by y = x, the height of the triangle is equal to x.
Step 4: Find the circumference of the shell at each x-value. The circumference of the shell is equal to the distance between the line x = 9 and the x-value. This distance is 9 - x.
Step 5: Set up the integral. The volume of the solid can be found by integrating the product of the height and circumference of each shell over the limits of integration. So the integral is:
V = ∫[from 0 to 9] of [(2πx)(9 - x)] dx
Step 6: Simplify and solve the integral. Expanding the expression and integrating, we have:
V = 2π ∫[from 0 to 9] of (9x - x^2) dx
= 2π [(4.5x^2 - (1/3)x^3)] [from 0 to 9]
= 2π [(4.5(9^2) - (1/3)(9^3)) - (4.5(0^2) - (1/3)(0^3))]
= 2π [(4.5(81) - (1/3)(729)) - (0)]
= 2π [(364.5 - 243) - (0)]
= 2π (121.5)
= 243π
Step 7: Simplify the result. The volume of the solid generated by revolving the region bounded by the given equations about the line x = 9 is 243π cubic units.
Therefore, the volume of the solid is 243π cubic units.