1. Given f(x) = √ x and g(x) = (x − 3)^2 ,

Question

(a)Plot both functions on the same axes, with x-interval [0, 5].

(b) Find the volume of the solid generated by rotating the region bounded by the two curves

around the line x = 0.5.

(c) Find the volume of the solid generated by rotating the region bounded by the two curves

around the line y = 5.

Answer 1

# a)

# import libaries

import matplotlib.pyplot as plt

import numpy as np

from scipy.integrate import quad

# Define the functions f(x) and g(x)

def f(x):

return np.sqrt(x)

def g(x):

return (x - 3) ** 2

# Generate x values between 0 and 5

x_values = np.linspace(0, 5, 400)

# Compute the corresponding y values for both functions

f_values = f(x_values)

g_values = g(x_values)

# Plot the functions

fig, ax = plt.subplots()

plt.figure(figsize=(8, 6))

ax.plot(x_values, f_values, label='f(x) = √x', color='blue')

ax.plot(x_values, g_values, label='g(x) = (x - 3)^2', color='red')

# Labeling the plot

ax.set_xlabel('x')

ax.set_ylabel('y')

ax.legend()

# b)

# The formula for the volume of a solid generated by rotating a region around a vertical line x=0.5 is given by the method of cylindrical shells.

# First we define a function for the shell integrand and then we integrate from 0 to 5.

# This involves integration so we use the scipy library (from scipy.integrate import quad)

# Set boundaries for the integration

a, b = 0, 5

# Define the integrand for cylindrical shells

def integrand_shell(x):

return 2 * np.pi * (x - 0.5) * (f(x) - g(x))

# Perform the integration

volume_shell = quad(integrand_shell, a, b)

print(f"Volume of solid rotating around x = 0.5: {volume_shell[0]}")

# c)

# We need to user the washer method for rotation around a horizontal line 𝑦=5

# First we define a function for the washer integrand and then we integrate from 0 to 5.

# This also involves integration so we use the scipy library (from scipy.integrate import quad)

# Define the integrand for the washer method

def integrand_washer(x):

return np.pi * ((5 - f(x))**2 - (5 - g(x))**2)

# Perform the integration

volume_washer = quad(integrand_washer, a, b)

print(f"Volume of solid rotating around y = 5: {volume_washer[0]}")