2. Use sympy to compute the exact volumes of the following solids S:

Question

(a) S rests on the xy-plane, and its base is the region bounded between f(x) = sin(π/4 · x)

and g(x) = cos(π/2 ·x)−1 on 0 ≤ x ≤ 4. Cross-sections of S perpendicular to the x-axis

are isosceles triangles with height sin3

(π/4 · x).

(b) Cross-sections of S perpendicular to the x-axis for values 0 ≤ x0 ≤ 2 are ellipses of the

form (x^(2))/(f(x_(0))^(2))+(y^(2))/(g(x_(0))^(2))=1

where f(x) = (1 − x)^2 and g(x) = ln(3)^−1 ln(x + 1).

Hint: The area of an ellipse is given by πab, where a, b are the lengths of the ellipse’s semi-major and semi-minor axes.

(c) S is a 4-dimensional hypersphere with radius 1; that is, cross-sections of S perpendicular to the x-axis are 3-dimensional spheres with radius √1 − x^2

Hint: The same method you used previously also works here; just add an extra dimension.

Answer 1

Hi Natalie,

These problems can all be solved using the same general approach:

1) find the cross-sectional area as a function of one variable (x)

2) integrate with respect to x over the indicated interval

a)

The area of a triangle is given by 1/2 * (base) *(height), so we need to find an expression for the base in terms of x alone. We can get the base by subtracting to get:

B(x) = f(x) - g(x)

The height h(x) is given, so we just need to plug in for our integral like this:

V = ∫ 1/2 B(x) * h(x) dx

with bounds of integration (0,4).

In python, assuming the sympy package is already installed, you can run the following commands to replicate the calculation above and get a symbolic answer as well as a numerical approximation if you like.

from sympy import symbols, sin, cos, integrate, pi, init_printing

# setup printing

init_printing()

# create symbol

x = symbols('x')

# given functions

f = sin(pi/4*x)

g = cos(pi/2*x) - 1

# constructed functions

b = f - g # traingle base

h = sin(pi/4*x) # traingle height

# define area function to use as integrand

a = b*h/2

# integrate

i = integrate(a, (x, 0, 4))

# report

display(i)

print(i.evalf())

b)

Again, our approach here is to build the cross-sectional area function which is given to us in the problem statement:

A(x) = πab

Since we know that the semimajor and semiminor axes will correspond to f(x) and g(x) based on the structure of the given ellipse equation, we can plug them in to find:

A(x) = π * (1-x)^2 * ln(x+1) / ln(3)

Now we simply integrate this with respect to x in the interval (0,2). In sympy, this can be done as follows:

from sympy import symbols, ln, pi, integrate, init_printing

# setup printing

init_printing()

# create symbol

x = symbols('x')

# given functions

f = (1-x)**2

g = ln(x+1)/ln(3)

# define area

a = pi * f * g

# integrate

i = integrate(a, (x, 0, 2))

# report

display(i)

print(i.evalf())

c)

Again, we find the cross-sectional area, which in this case is a 3-dimensional volume, but the concept still applies. The volume of a sphere is given by:

V_sphere = (4/3)πr³

with r given by:

r = √(1-x²)

Now integrate with respect to x over the interval (-1,1).