Unit 3: Geometrical Applications of Integration

Plane Curves (Cartesian & Parametric)

This unit applies integration to curves defined in two primary ways:

1. Cartesian Equation: The relationship between x and y is given directly, usually as y = f(x) or x = g(y).
   Example: The parabola y = x².
2. Parametric Equation: Both x and y are defined as separate functions of a third variable (a parameter), usually t or θ.
   x = f(t), y = g(t)
   Example: The circle x = a cos(t), y = a sin(t).

For parametric equations, we often need the derivative dy/dx, which is found using the chain rule:

dy/dx = (dy/dt) / (dx/dt)

Rectification of Plane Curves (Arc Length)

Rectification is the process of finding the length of a curve, also known as its **arc length**.

Arc Length Element (ds)

We start by finding the length of an infinitesimally small segment of the curve, ds. By the Pythagorean theorem:

ds² = dx² + dy²

We can factor this in different ways:

• ds = sqrt(dx² + dy²)
• ds = sqrt(1 + (dy/dx)²) dx
• ds = sqrt((dx/dy)² + 1) dy
• ds = sqrt((dx/dt)² + (dy/dt)²) dt (for parametric)

To find the total arc length L, we integrate this ds element over the desired interval.

Cartesian Formula

To find the arc length of y = f(x) from x = a to x = b:

L = ∫ [from a to b] sqrt(1 + (dy/dx)²) dx

To find the arc length of x = g(y) from y = c to y = d:

L = ∫ [from c to d] sqrt(1 + (dx/dy)²) dy

Parametric Formula

To find the arc length of x = f(t), y = g(t) from t = t₁ to t = t₂:

L = ∫ [from t₁ to t₂] sqrt((dx/dt)² + (dy/dt)²) dt

Example: Find the circumference of a circle x = a cos(t), y = a sin(t) from t = 0 to t = 2π.

1. Find derivatives:
   dx/dt = -a sin(t)
dy/dt = a cos(t)
2. Square and add:
   (dx/dt)² + (dy/dt)² = (-a sin(t))² + (a cos(t))²
= a²sin²(t) + a²cos²(t) = a²(sin²(t) + cos²(t)) = a²
3. Set up the integral:
   L = ∫ [from 0 to 2π] sqrt(a²) dt = ∫ [from 0 to 2π] a dt
4. Evaluate:
   L = a [t] from 0 to 2π = a(2π - 0) = 2πa.

Areas of Surfaces of Revolution

This is the area of the 3D surface created when a plane curve is rotated around an axis. The general formula involves integrating the circumference of a small "strip" (2π * radius) along the arc length (ds).

Area = ∫ 2π · (radius) · ds

Revolution about the x-axis

When rotating around the x-axis, the radius of rotation for any point (x, y) is its y-coordinate.

• Cartesian: Aₓ = ∫ [from a to b] 2πy ds = ∫ [from a to b] 2πy · sqrt(1 + (dy/dx)²) dx
• Parametric: Aₓ = ∫ [from t₁ to t₂] 2πy(t) · sqrt((dx/dt)² + (dy/dt)²) dt

Revolution about the y-axis

When rotating around the y-axis, the radius of rotation for any point (x, y) is its x-coordinate.

• Cartesian: Aᵧ = ∫ 2πx ds = ∫ [from a to b] 2πx · sqrt(1 + (dy/dx)²) dx
• Parametric: Aᵧ = ∫ [from t₁ to t₂] 2πx(t) · sqrt((dx/dt)² + (dy/dt)²) dt

Volumes of Solids of Revolution

This is the volume of the 3D solid created when a 2D area (under a curve) is rotated around an axis.

Disk Method (about x-axis)

We find the volume by summing an infinite number of thin "disks" (cylinders) of radius y and thickness dx. The volume of each disk is dV = (Area of base) · (height) = (πy²) · dx.

Vₓ = ∫ [from a to b] πy² dx = ∫ [from a to b] π[f(x)]² dx

Parametric: Vₓ = ∫ [from t₁ to t₂] π[y(t)]² · (dx/dt) dt

Disk Method (about y-axis)

Similarly, when rotating around the y-axis, the disks have radius x and thickness dy. The volume of each disk is dV = (πx²) · dy.

Vᵧ = ∫ [from c to d] πx² dy = ∫ [from c to d] π[g(y)]² dy

Parametric: Vᵧ = ∫ [from t₁ to t₂] π[x(t)]² · (dy/dt) dt

Example: Find the volume of a sphere of radius r.

1. We can rotate the semi-circle y = sqrt(r² - x²) around the x-axis, from x = -r to x = r.
2. Use the Disk Method: Vₓ = ∫ [from -r to r] πy² dx
3. Substitute y² = r² - x²:
   V = ∫ [from -r to r] π(r² - x²) dx
4. Since (r² - x²) is an even function, we can use Property P-7:
   V = 2π ∫ [from 0 to r] (r² - x²) dx
5. Integrate:
   V = 2π [r²x - x³/3] from 0 to r
V = 2π [(r³ - r³/3) - (0 - 0)]
V = 2π [2r³/3] = (4/3)πr³.

Unit 3: Exam Quick Tips