Answer:
V = 81π/4
To find the volume of the solid formed by rotating the region bounded by the graph of y = x^2, x = 3, and y = 0 around the x-axis, we can use the method of cylindrical shells.
The formula for the volume using cylindrical shells is:
V = ∫[a,b] 2πx * f(x) * dx
In this case, the region is bounded by x = 3, y = 0, and the curve y = x^2. We need to determine the limits of integration (a and b) based on the intersection points of these curves.
Since y = x^2 and y = 0, we can set x^2 = 0 and solve for x to find the intersection point:
x^2 = 0
x = 0
Therefore, the limits of integration are from x = 0 to x = 3.
Now we can set up the integral:
V = ∫[0,3] 2πx * (x^2) * dx
Integrating this expression will give us the volume of the solid. Evaluating the integral:
V = ∫[0,3] 2πx^3 dx
V = π * [(1/4)x^4] [0,3]
V = π * (1/4) * (3^4 - 0^4)
V = π * (1/4) * (81 - 0)
V = π * (1/4) * 81
V = 81π/4
Therefore, the volume of the solid formed by rotating the region bounded by the graph of y = x^2, x = 3, and y = 0 around the x-axis is 81π/4 cubic units.