Multiple integrals and applications notes — Unit 4

A single integral ∫ f(x) dx finds the area under a 2D curve. But what if you have a 3D surface z = f(x,y) and you want to find the volume underneath it? You need a Double Integral: ∬ f(x,y) dx dy. And what if you want to find the mass of a solid block where the density varies at every point? You need a Triple Integral: ∭ f(x,y,z) dx dy dz.

⇒The Geometry of Limits

In a single integral, limits are just two numbers on the x-axis: x=a to x=b. In a double integral, the limits define a 2D region (an area) on the xy-plane.

To evaluate ∬ f(x,y) dx dy, we work from the INSIDE OUT.

$Evaluate I = int(x=0 to 1)$ $int(y=0 to x) (x$^2 + y$^2) dy dx$

1. 1.Step 1: The inner integral is w.r.t 'y' (from y=0 to y=x). Treat x as a constant.
2. 2.$Inner Integral = int (x$^2 + y$^2) dy$
3. 3. $= [ x^2y + y$^3/3 ] evaluated from y=0 to y=x$
4. 4. $= ( x^2(x) + x$^3/3 ) - ( 0 + 0 )$
5. 5. $= x^3 + x$^3/3$
6. 6. $= (4/3)x^3$

Often, you are not given the explicit limits. You are told: '$Integrate f(x,y) over the region bounded by y=x and y=x^2$'. You must sketch the region and extract the limits yourself.

Always draw a tiny representative strip (rectangle) on your sketch. This single habit prevents 90% of exam errors.

This is the most guaranteed 10-mark question in Unit 4. You will be given an integral that is mathematically impossible to evaluate as written (e.g.$, integrating e^(y$^2) dy)$. The ONLY way to solve it is to change the order from dy dx to dx dy.

Change the order and evaluate: I = ∫(y=0 to ∞) ∫(x=y to ∞) [ e^(-x) / x ] dx dy

1. 1.Attempting to evaluate directly: ∫ e^(-x)/x dx has no closed-form elementary anti-derivative. It's impossible. We must change order.
2. 2.Step 1: Extract old limits.
3. 3.Inner (dx): x varies from x = y to x = ∞
4. 4.Outer (dy): y varies from y = 0 to y = ∞

Step 3: Rotate the strip.

1. 1.Step 4: Extract new limits.
2. 2.Inner (dy): y varies from y = 0 to y = x.
3. 3.Outer (dx): To cover the whole infinite wedge, x must scan from x = 0 all the way to x = ∞.

Now let's evaluate the newly ordered integral.

1. 1.This is amazing. The annoying 'x' in the denominator completely canceled out with the 'x' limit from the inner integration.
2. 2.Step 2: Evaluate outer integral w.r.t 'x'.
3. 3.I = ∫(x=0 to ∞) e^(-x) dx
4. 4. = [ -e^(-x) ] from 0 to ∞
5. 5. = ( -e^(-∞) ) - ( -e^(0) )
6. 6. = ( 0 ) - ( -1 )
7. 7. = 1

Sometimes Cartesian (x,y) coordinates are terrible for the region. $If the region is a circle x^2 + y$^2 = a$^2, vertical strips will have limits involving square roots$: y = ±sqrt{a^2 - x^2}. Integrating this is brutal. $We switch to Polar Coordinates (r, heta)$.

Evaluate ∬ $e^(-(x$^2+y$^2)) dx dy over the positive quadrant of a circle of radius R$.

1. 1.1. Transform variables: $x^2 + y$^2 becomes r$^2$.
2. 2.2. Transform differential: dx dy becomes r dr dθ.
3. 3.3. Transform region limits:
4. 4. The region is a quarter circle (first quadrant).
5. 5. r (distance from origin) goes from 0 to R.
6. 6. θ (angle from x-axis) goes from 0 to π/2.

Triple Integrals ∭ f(x,y,z) dx dy dz follow the exact same logic as double integrals. Work inside out. The innermost integral has limits depending on 2 variables. The middle integral has limits depending on 1 variable. The outermost integral has constant limits.

Always look at the region of integration. $If it has x^2 + y$^2 + z$^2 in it, switch to spherical immediately$. $If it has x^2 + y$^2, switch to cylindrical$.

Just like dx dy = r dr dθ required a Jacobian, switching a triple integral requires a 3x3 Jacobian.

To find the pure area of a 2D region, simply evaluate the double integral with f(x,y) = 1.

1. 1.Example: $Find area between parabola y = x^2 and line y = x$.
2. 2.1. Find intersections: $x^2 = x =$> x(x-1) = 0 => x=0, x=1.
3. 3.2. Set limits: x goes 0 to 1. $For a given x, y goes from x^2 (bottom curve) to x (top curve)$.
4. 4.3. $Area = int(x=0 to 1)$ $int(y=x$^2 to x) 1 dy dx$
5. 5.4. Inner: $[y] from x^2 to x = (x - x$^2)$
6. 6.5. Outer: $int(x-x$^2) dx = [x$^2/2 - x$^3/3] from 0 to 1 = 1/2 - 1/3 = 1/6$.

To find the pure volume of a 3D region, simply evaluate the triple integral with f(x,y,z) = 1.

Alternatively, if you know the ceiling function z = f(x,y) and the floor is z=0, you can use a double integral.

$Prove the volume of a sphere of radius R is (4/3)piR$^3$.

1. 1.Set limits for a full sphere:
2. 2.r (radius) goes from 0 to R
3. 3.θ (latitude) goes from 0 to π (North pole to South pole)
4. 4.φ (longitude) goes from 0 to 2π (Full circle around)

$If a solid has a non-uniform density, say ho(x,y,z) = x$^2yz, you cannot just say Mass = Volume * Density$. You must integrate the density over the volume.

These are long, tedious calculations. In exams, look for symmetry! If the object is perfectly symmetric across the yz-plane, and the density is symmetric, then x̄ = 0 instantly by inspection.

Dirichlet's Theorem is a powerful shortcut for a very specific type of triple integral that appears constantly in exams.

Find the mass of a solid in the first octant bounded by x+y+z=1, if density ρ = xyz.

1. 1.Step 1: Set up the integral.
2. 2.Mass = ∭ (xyz) dx dy dz
3. 3.Step 2: Match to Dirichlet form: x^(l-1) y^(m-1) z^(n-1)
4. 4.$Here, x = x^1$. So l-1 = 1 => l=2.
5. 5.Similarly, m=2, n=2.

We solved a mass integration problem over a tetrahedral region in 3 lines of algebra without doing a single integration step. This is the power of Dirichlet.

What if the region is bounded by (x/a)^p + (y/b)^q + (z/c)^r ≤ 1? Dirichlet's standard theorem fails because of the powers and constants.

1. 1.Liouville allows us to use substitution to 'force' it into Dirichlet form.
2. 2.Let u = (x/a)^p => x = a * u^(1/p)
3. 3.Then dx = (a/p) * u^(1/p - 1) du
4. 4.Do the same for y (into v) and z (into w).
5. 5.The region is now strictly u + v + w ≤ 1.

⇒Unit 4 Mastery

• Can you draw horizontal and vertical strips to extract limits from a 2D sketch?
• Do you remember the 4-step algorithm to Change the Order of Integration?
• Can you correctly evaluate a double integral from the INSIDE OUT?
• $Do you remember the Jacobians for Polar (r), Cylindrical (r), and Spherical (r^2sin$ heta)$?
• Can you set up a triple integral to find Volume and Mass?
• Can you execute Dirichlet's formula for tetrahedral boundary conditions?