Unit 5: Geometrical Applications of Integration

Area Bounded by Plane Curves

Cartesian Coordinates

To find the area of a region, we "slice" it into an infinite number of thin rectangles and sum their areas using integration.

• Area with respect to x-axis:
  For a region bounded by y = f(x), the x-axis (y=0), x = a, and x = b.
  Area of one rectangle: dA = y dx

  Area = ∫ [from a to b] y dx = ∫ [from a to b] f(x) dx

• Area with respect to y-axis:
  For a region bounded by x = g(y), the y-axis (x=0), y = c, and y = d.
  Area of one rectangle: dA = x dy

  Area = ∫ [from c to d] x dy = ∫ [from c to d] g(y) dy

• Area between two curves:
  For a region bounded above by y = f(x) and below by y = g(x) from x = a to x = b.
  Height of rectangle: (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

Polar Coordinates

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

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Rectification of Plane Curves (Arc Length)

Rectification is the process of finding the length of a curve (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 (Cartesian)
ds² = (dr)² + (r dθ)² (Polar)

We can factor these to get the element we integrate:

• Cartesian: ds = sqrt(1 + (dy/dx)²) dx
• Polar: ds = sqrt(r² + (dr/dθ)²) dθ

Cartesian Formula

To 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

Polar Formula

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

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Volumes of Solids of Revolution (Cartesian)

This is the volume of the 3D solid created when a 2D area is rotated around an axis. The syllabus specifies Cartesian curves.

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

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

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Surfaces of Solids of Revolution (Cartesian)

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

Revolution about the x-axis

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

Revolution about the y-axis

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

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Unit 5: Exam Quick Tips