Question

define q as the region bounded by the functions f(x)=x24 and g(x)=2x in the first quadrant between y=2 and y=4. if q is rotated around the y-axis, what is the volume of the resulting solid?

Answer 1

To find the volume of the solid formed by rotating region q around the y-axis, we can use the disk method. The volume can be calculated using the formula V = π ∫[a, b] (f(x)^2 - g(x)^2) dx. Therefore, the volume is 8π.

Step-by-step explanation:

To find the volume of the solid formed by rotating region q around the y-axis, we can use the disk method. Since q is bounded by the functions f(x) = x^2 and g(x) = 2x, we need to determine the limits of integration. Setting the equations equal to each other, x^2 = 2x, we find x = 0 and x = 2 are the x-coordinates of where the two curves intersect. The limits of integration for the y-values are 2 and 4.

Using the formula for the volume of a solid of revolution, V = π ∫[a, b] (f(x)^2 - g(x)^2) dx, we can substitute in the given functions and limits of integration to calculate the volume.

Therefore, the volume of the resulting solid is (4π/3)((2^3 - 2^4) - (0^3 - 0^4)) = 24π/3 = 8π.