Final answer:
To find the volume of the solid obtained by rotating a region bounded by y = 1/36x^2, x = 5, and y = 0 about the y-axis, we use the method of washers. Expressing x in terms of y, we integrated the area function A(y) = πx^2 from y=0 to y=25/36, finally yielding a volume of 31.25π cubic units.
Step-by-step explanation:
To find the volume V of a solid obtained by rotating the region bounded by the curves given by y = 1/36x^2, x = 5, and y = 0; about the y-axis, we will use the method of disks or washers. This method involves integrating across the given boundary of the solid's cross-section perpendicular to the axis of rotation.
The area function A(y) of a washer is the area of the outer disk minus the area of the inner disk. Since the region is rotated about the y-axis and bounded by x = 5, we only consider the area of the outer disk which is defined by the curve y = 1/36x^2, and since we have no inner disk (the y-axis is the boundary on the other side), the area function becomes the area of a circle with radius r = x. To calculate this, we must express x in terms of y because we are integrating with respect to y. Solving for x in the equation y = 1/36x^2, we get x = √(36y). So the area function is A(y) = πx^2 = π(36y).
The boundaries of y are from 0 (where y = 0) to a maximum y value that occurs at x = 5; substituting into y = 1/36x^2, we get y = 25/36. Therefore, we are integrating A(y) from 0 to 25/36 in terms of y.
The volume V is thus the integral of A(y) from 0 to 25/36:
V = ∫025/36 π(36y) dy
Computing this integral, we have:
V = π ∫025/36 36y dy = π[18y^2]025/36 = π(18 · (25/36)^2) = π(18 · 625/1296) = π1250π/36 = 31.25π
The volume of the solid obtained by rotating the region about the y-axis is 31.25π cubic units.