Final answer:
To find the volume of the solid obtained by rotating the region enclosed by the graphs of y=15-x and y=3x-5 about the y-axis, we can use the method of cylindrical shells. The volume is approximately 62.83 cubic units.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region enclosed by the graphs of y=15-x, y=3x-5, and x=0 about the y-axis, we can use the method of cylindrical shells. First, we need to find the limits of integration, which are the x-values where the two curves intersect. Setting the two equations equal to each other, we get:
15-x = 3x-5
8x = 20
x = 2.5
So the limits of integration are from x=0 to x=2.5. Next, we find the radius of each cylindrical shell at each x-value by subtracting the y-values of the curves:
r = (15 - x) - (3x - 5) = 20 - 4x
Finally, we calculate the volume using the formula for cylindrical shells:
V = 2π ∫(0 to 2.5) [ (20 - 4x) * x ] dx
After evaluating the integral, the volume is:
V ≈ 62.83 cubic units.