2-3. Integration

Question 1

How do we write the integral for arbitrary (n) dimensions?

Answer 1

Keep in mind that there is always only one integral sign, and that x is a vector.

Question 2

In general, what is the notation for the volume of a domain, and how is it computed?

Answer 2

Integrate f(x) = 1 over the domain.

Question 3

If you multiply every point in some domain by a constant, what happens to the volume of the domain?

Answer 3

The volume is multiplied by the constant to the power of the dimension:

vol_n(cD) = cⁿvol_n(D)

We can call this the “volume-scaling formula”.

Question 4

Explain the process of finding the volume of the n-dimensional unit sphere.

Answer 4

1. Observe that the sum of the squares of all the coordinates must be less than 1 (due to the distance formula).
2. Now, if we fix the last of these coordinates, x_n, and subtract it, the sum of the squares of all the coordinates up to x_n-1 must be less than 1-x_n²
3. Now we have a sphere in n-1 dimensions where the radius squared is equal to 1-x_n². If we vary x_n from -1 to 1 and integrate that, then we will have the volume of the n-dimensional sphere.
4. We can use the volume scaling formula to convert the n-1-dimensional ball’s radius to 1.
5. Now, since the volume of the n-1-dimensional unit ball is independent of our integration variable x_n, pull it out of the integral. Now you have a recurrent definition for the volume of an n-dimensional unit sphere based on the volume of an n-1 dimensional unit sphere.

Question 5

Give two examples of “simplices” (plural for the term “simplex”).

Answer 5

A triangle and a tetrahedron are both simplices.

Question 6

What defines a unit simplex in n-dimensions?

Answer 6

A unit simplex is always defined by the sum of all the coordinates x¹ + x² + … + xⁿ being less than or equal to one.

Question 7

In general, what occurs when you take an n-dimensional object and fix one of the coordinates?

Answer 7

The shape defined by the remaining variables becomes an n-1-dimensional object of the same kind, but with a different scale (the scale depends on what value you fix xⁿ at).

Question 8

What do you have to remember when doing trig-substitution to solve an integral?

Answer 8

• Remember to convert the bounds of the integral to the new variable (theta) using your substitution equation.
• Remember to convert the differential of integration to the new variable by taking the derivative of your substitution equation
• If you have converted the function, the bounds, and the differential of integration, you are free to integrate!! (This is true for any substitution method)

Question 9

What is the formula of the volume of the unit n-simplex?

Answer 9

1/n!