Question

define q as the region bounded by the function f(x)=x and the line x=1 in the first quadrant between y=1 and y=3. if q is rotated around the y-axis, what is the volume of the resulting solid?

Answer 1

The volume of the solid obtained by rotating the region q, bounded by the function f(x) = x and the line x = 1 between y=1 and y=3, when rotated around the y-axis, is calculated using disk integration methods yielding a volume of 2π cubic units.

Step-by-step explanation:

To find the volume of the resulting solid when the region q is rotated around the y-axis, we use the method of disk integration, which is a part of the calculus subject. The region q is bounded by the function f(x) = x and the line x = 1 between y = 1 and y = 3 in the first quadrant. To set up the integral, we consider the disks with radius r(y) = 1 since x = 1 forms the outer boundary of the region q as we revolve around the y-axis.

The volume (V) of the solid can be found by integrating the area of each disk as a function of y:

V = π ∫ (r(y))^2 dy

Here the radius is constant, and the limits of integration are from y = 1 to y = 3:

V = π ∫_{1}^{3} (1)^2 dy = π(y)|_{1}^{3} = π(3 - 1)

The result is V = 2π, which is the volume of the cylindrical shell obtained by rotating the region q around the y-axis.