How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #125 y = x^3# , y = 8 , x = 0 revolved about the x-axis?

Question

Cameron Alexander · Accepted Answer

See below

Here is the region:

A representative slice taken perpendicular to the axis of rotation has volume

#int \ pi(8^2 - (x^3/5^3)^2) dx# using #VOR = int pi y^2 dx#

#x# varies from #0# to #10#, so the solid will have volume:

#piint_0^10 (64 - x^6/5^6) dx = pi[64x-x^7/(7(5^6))]_0^10#

# = pi[640 - 10^7/(7(5^6))]#

# = pi[640 - (10 * 2^6)/7]#
# = pi[640 - 640/7]#
# = pi[(7*640)/7 - 640/7]#
# = pi (6 * 640)/7#
# = (3840pi)/7#
We can also use cylindrical shells to get the same answer:

# V =int_(y=a)^(y=b)2pi \ y \ g(y) \ dy #
# \ \ =2pi int_0^8 \ y \ root(3)(125y) \ dy #
# \ \ =2pi int_0^8 5 \ y \ y^(1/3) \ dy #
# \ \ =10pi int_0^8 y^(4/3) \ dy #
# \ \ =10pi [y^(7/3)/(7/3)]_0^8 #
# \ \ =(30pi)/7 [y^(7/3)]_0^8 #
# \ \ =(30pi)/7 (128-0) #
# \ \ =(3840pi)/7 #

Samantha Adkison · Answer

undefined To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by $ y = x^3 $, $ y = 8 $, and $ x = 0 $ about the x-axis, follow these steps:

1. Determine the limits of integration. In this case, the limits of integration for x are from 0 to the x-coordinate where $ y = x^3 $ intersects $ y = 8 $. Solving $ x^3 = 8 $ gives $ x = 2 $.

2. Determine the radius of each cylindrical shell. The radius of each shell is the distance from the axis of rotation (the x-axis) to the curve $ y = x^3 $ at a particular value of x. So, the radius $ r $ is $ x $.

3. Determine the height of each cylindrical shell. The height of each shell is the difference between the upper and lower functions at a particular x-value. Here, the upper function is $ y = 8 $, and the lower function is $ y = x^3 $. So, the height $ h $ is $ 8 - x^3 $.

4. Write the expression for the volume of each cylindrical shell. The volume $ V $ of a cylindrical shell is given by $ V = 2\pi rh $, where $ r $ is the radius and $ h $ is the height.

5. Integrate the volume expression with respect to $ x $ from 0 to 2.

$$ V = \int_{0}^{2} 2\pi x(8 - x^3) \, dx $$

6. Evaluate the integral to find the volume.

$$ V = 2\pi \int_{0}^{2} (8x - x^4) \, dx $$

$$ = 2\pi \left[ 4x^2 - \frac{x^5}{5} \right]_{0}^{2} $$

$$ = 2\pi \left( 4(2)^2 - \frac{(2)^5}{5} \right) - 2\pi \left( 4(0)^2 - \frac{(0)^5}{5} \right) $$

$$ = 2\pi \left( 16 - \frac{32}{5} \right) - 0 $$

$$ = 2\pi \left( \frac{80}{5} - \frac{32}{5} \right) $$

$$ = 2\pi \left( \frac{48}{5} \right) $$

$$ = \frac{96\pi}{5} $$

Therefore, the volume of the solid obtained by rotating the given region about the x-axis using the method of cylindrical shells is $ \frac{96\pi}{5} $ cubic units.