Question

Find the volume of the solid formed by rotating the region enclosed by

x=0, x=1, y=0, y=9+x7
about the x-axis.
V=_____ cubic units

Answer 1

Explanation:

To find the volume of the solid formed by rotating the region about the x-axis, we can use the method of disks.

At a given value of x, the distance between the curve y = 9 + x^2 and the x-axis is 9 + x^2. Thus, the area of the disk at x is A(x) = π(9 + x^2)^2. The limits of integration are 0 and 1, since the region is bounded by the lines x = 0 and x = 1.

Therefore, the volume of the solid is given by:

V = ∫(0 to 1) π(9 + x^2)^2 dx

Using integration techniques (such as substitution), we can evaluate this integral to get:

V = (112π/5) cubic units (rounded to 3 decimal places)

Therefore, the volume of the solid formed by rotating the region about the x-axis is (112π/5) cubic units