Answer based on the code above:

import numpy as np import matplotlib.pyplot as plt from mpl$\_$toolkits.mplot$3$d import Axes$3$D # Define the function f$($x$)$ that models the vase's profile def vase$\_$profile$($x$)$: # Example: a quadratic profile with a flared top return $1+0\mathrm{.}5*$ x$**2$ # f$($x$)=1+0\mathrm{.}5$x$^2$ # Set up the range of x $($e$.$g$.,$ height of the vase, in cm$)$ x $=$ np$.$linspace$(-2,2,500)$ # x ranges from $-2$ cm to $2$ cm y $=$ vase$\_$profile$($x$)$ # Plot the $2$D profile of the vase plt$.$figure$($figsize$=(6,6))$ plt$.$plot$($x$,$ y$,$ label$=$'Vase Profile $($f$($x$))\text{'},$ color$=$'blue'$)$ plt$.$title$("2$D Profile of the Vase"$)$ plt$.$xlabel$("$Radius $($cm$)")$ plt$.$ylabel$("$Height $($cm$)")$ plt$.$axhline$(0,$ color$=$'black', linewidth$=0\mathrm{.}5)$ plt$.$axvline$(0,$ color$=$'black', linewidth$=0\mathrm{.}5)$ plt$.$legend$()$ plt$.$grid$()$ plt$.$show$()$ # Generate the solid of revolution X $=$ np$.$linspace$(-2,2,500)$ Y $=$ vase$\_$profile$($X$)$ theta $=$ np$.$linspace$(0,2*$ np$.$pi$,500)$ # Revolve around y$-$axis X$,$ Theta $=$ np$.$meshgrid$($X$,$ theta$)$ Y $=$ vase$\_$profile$($X$)$ Z $=$ X $*$ np$.$cos$($Theta$)$ R $=$ X $*$ np$.$sin$($Theta$)$ # $3$D plot of the solid of revolution fig $=$ plt$.$figure$($figsize$=(10,8))$ ax $=$ fig.add$\_$subplot$(111,$ projection$=\text{'}3$d$\text{'})$ ax$.$plot$\_$surface$($Z$,$ R$,$ Y$,$ cmap$=$'viridis', edgecolor$=\text{'}$k$\text{'},$ alpha$=0\mathrm{.}8)$ ax$.$set$\_$title$("$Solid of Revolution of the Vase"$)$ ax$.$set$\_$xlabel$("$X $($cm$)")$ ax$.$set$\_$ylabel$("$Z $($cm$)")$ ax$.$set$\_$zlabel$("$Height $($cm$)")$ plt$.$show$()$

$($e$)$ Use either of the frameworks developed above to find an estimate for the volume

of your object. Discuss the method that you used, justifying your choice.

$($f$)$ Create a visualization of your chosen method.

$($g$)$ Estimate the total volume of your object and compare your measurement with

your integral calculations. Discuss any discrepancies between the two.