To find the area of a region, we "slice" it into an infinite number of thin rectangles and sum their areas using integration.
y = f(x), the x-axis (y=0), x = a, and x = b.
dA = y dx
Area = ∫ [from a to b] y dx = ∫ [from a to b] f(x) dx
x = g(y), the y-axis (x=0), y = c, and y = d.
dA = x dy
Area = ∫ [from c to d] x dy = ∫ [from c to d] g(y) dy
y = f(x) and below by y = g(x) from x = a to x = b.
(y_upper - y_lower) = f(x) - g(x)
Area = ∫ [from a to b] (y_upper - y_lower) dx = ∫ [from a to b] [f(x) - g(x)] dx
For a polar curve r = f(θ), we find the area by summing an infinite number of thin "sectors" (like pizza slices).
The area of a small sector of a circle is dA = (1/2)r² dθ.
To find the total area swept by the radius vector from θ = α to θ = β:
Area = ∫ [from α to β] (1/2)r² dθ = (1/2) ∫ [from α to β] [f(θ)]² dθ
Rectification is the process of finding the length of a curve (its arc length).
We start by finding the length of an infinitesimally small segment of the curve, ds. By the Pythagorean theorem:
ds² = dx² + dy² (Cartesian)
ds² = (dr)² + (r dθ)² (Polar)
We can factor these to get the element we integrate:
ds = sqrt(1 + (dy/dx)²) dxds = sqrt(r² + (dr/dθ)²) dθTo find the arc length L of y = f(x) from x = a to x = b:
L = ∫ [from a to b] ds = ∫ [from a to b] sqrt(1 + (dy/dx)²) dx
If the curve is given as x = g(y) from y = c to y = d:
L = ∫ [from c to d] sqrt(1 + (dx/dy)²) dy
To find the arc length L of r = f(θ) from θ = α to θ = β:
L = ∫ [from α to β] ds = ∫ [from α to β] sqrt(r² + (dr/dθ)²) dθ
This is the volume of the 3D solid created when a 2D area is rotated around an axis. The syllabus specifies Cartesian curves.
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
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
This is the surface area of the 3D solid created when a 2D curve (an arc) 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
When rotating around the x-axis, the radius of rotation for any point (x, y) is its y-coordinate.
Aₓ = ∫ 2πy ds = ∫ [from a to b] 2πy · sqrt(1 + (dy/dx)²) dx
When rotating around the y-axis, the radius of rotation for any point (x, y) is its x-coordinate.
Aᵧ = ∫ 2πx ds = ∫ [from a to b] 2πx · sqrt(1 + (dy/dx)²) dx
Do NOT confuse the formulas! This is the most common mistake.
| Application | Key Element (Cartesian) | Key Element (Polar) |
|---|---|---|
| Area | y dx |
(1/2)r² dθ |
| Arc Length (L) | ds = sqrt(1 + (y')²) dx |
ds = sqrt(r² + (r')²) dθ |
| Surface Area (Aₓ) (about x-axis) |
2πy ds |
2π(r sinθ) ds (ds is polar) |
| Volume (Vₓ) (about x-axis) |
πy² dx |
(Not typically done this way) |
∫ y dx. Volume is ∫ πy² dx. Don't forget the π or the square for volume.ds. Surface Area is just ds multiplied by the circumference 2π * radius.(1/2)r² dθ.sqrt(r² + (dr/dθ)²) dθ. They are very different.