Find the volumes of the solids generated by revolving the region in the first quadrant bounded by the following about the given axes.
x = y - y3, x = 1, y = 1
(b) Revolved about the y-axis ?
Hi Shawn,
97/105 * pi ?
:)
We need to cut away the volume of the region you get by revolving the area inbetween the y-axis and the line x = y - y^3 from the "total volume" you get by revolving the entire square bounded by the x-axis, the y-axis the line x = 1 and the line y = 1. The latter is pi. The former is:
pi * Integral from zero to 1 of dy (y - y^3)^2 =
pi*(1/3 - 2/5 + 1/7) = 8/105*pi
The volume is thus pi - 8/105*pi = 97/105 * pi
ggggggggggggggggggggggggggggggggggggggggggggggggggggggggggg
Anonymous,this is not a social site.This is for help not for you to post gggggggggggggggggggggggggg repeatedly.
To find the volume of the solid generated by revolving the region in the first quadrant bounded by the curves x = y - y^3, x = 1, and y = 1 about the y-axis, you can follow these steps:
1. First, find the limits of integration for the y-variable. In this case, the region is bounded by y = 0 (the x-axis) and y = 1. So the limits of integration will be from y = 0 to y = 1.
2. Set up the integral for the volume using the disk method. The volume of each infinitesimally thin disk can be calculated using the formula V = π(radius)^2(height), where the radius is the x-value and the height is the differential dy.
3. The radius of each disk is the distance from the y-axis to the curve x = y - y^3. So the radius, in this case, is x = y - y^3.
4. The height of each disk is the differential dy.
5. Square the equation for x to get rid of the square root: x^2 = (y - y^3)^2.
6. Set up and evaluate the integral: V = π * ∫(from 0 to 1) (y - y^3)^2 dy.
7. Integrate the function using the power rule for integration and evaluate it over the limits of integration.
8. Subtract the integral value from the total volume of the square. The total volume of the square is given by the formula V_square = π * base * height, which in this case is π * 1 * 1 = π.
9. Plug in the calculated value into the formula V_total - V_cut = V_final, where V_total is the total volume of the square, V_cut is the cut-away volume, and V_final is the final volume of the solid.
Following these steps, you will find that the volume of the solid generated by revolving the region about the y-axis is 97/105 * π.