Unit 5: Applications of Integration

Table of Contents

Area Bounded by Plane Curves (Cartesian)

Integration can be used to find the area of a region bounded by one or more curves.

Area under a curve y = f(x) from x = a to x = b

If f(x) ≥ 0 on [a, b], the area A is the integral of the function.

A = ∫ [a to b] f(x) dx

Area between two curves y = f(x) and y = g(x)

If f(x) ≥ g(x) on [a, b], the area A between them is the integral of the "upper curve" minus the "lower curve".

A = ∫ [a to b] [f(x) - g(x)] dx
Exam Tip:
  1. Sketch the curves! This is the most important step. You need to see which function is on top.
  2. Find the points of intersection by setting f(x) = g(x) and solving for x. These will be your limits of integration (a and b).
  3. Set up the integral: ∫ [a to b] (y_upper - y_lower) dx.
  4. If the "upper" curve changes, you must split the integral into multiple parts.

Integration with respect to y

If the region is bounded by curves x = f(y) and x = g(y) from y = c to y = d, it's easier to integrate with respect to y.

If f(y) ≥ g(y) (meaning f(y) is the "right" curve and g(y) is the "left" curve):

A = ∫ [c to d] [f(y) - g(y)] dy = ∫ [c to d] (x_right - x_left) dy

Area Bounded by Plane Curves (Polar)

For a curve given in polar coordinates r = f(θ), the area is found by integrating "sectors" instead of rectangles.

Area of a Polar Region

The area A of the region bounded by the curve r = f(θ) and the rays θ = α and θ = β is:

A = (1/2) * ∫ [α to β] r² dθ = (1/2) * ∫ [α to β] [f(θ)]² dθ

Area Between Two Polar Curves

If the region is bounded by two curves r = f(θ) (outer curve) and r = g(θ) (inner curve) from θ = α to θ = β:

A = (1/2) * ∫ [α to β] ( [f(θ)]² - [g(θ)]² ) dθ = (1/2) * ∫ [α to β] (r_outer² - r_inner²) dθ
Common Mistake: Do NOT integrate (r_outer - r_inner)². The correct formula is (r_outer² - r_inner²).
Tip: Finding the limits α and β often involves setting f(θ) = g(θ) or finding where r = 0 (the pole). Symmetry is very helpful (e.g., find area of one loop and multiply).

Rectification of Plane Curves (Cartesian)

Rectification means finding the length of an arc of a curve.

If s is the arc length of a curve y = f(x) from x = a to x = b, the formula is:

s = ∫ [a to b] √(1 + (dy/dx)²) dx

If the curve is given as x = g(y) from y = c to y = d:

s = ∫ [c to d] √(1 + (dx/dy)²) dy

If the curve is given in parametric form x = x(t), y = y(t) from t = t₁ to t = t₂:

s = ∫ [t₁ to t₂] √((dx/dt)² + (dy/dt)²) dt

Rectification of Plane Curves (Polar)

For a curve given in polar coordinates r = f(θ) from θ = α to θ = β, the arc length s is:

s = ∫ [α to β] √(r² + (dr/dθ)²) dθ

Example: Find the length of the cardioid r = a(1 + cos θ).

  1. dr/dθ = -a sin θ
  2. r² + (dr/dθ)² = [a(1 + cos θ)]² + [-a sin θ]²
    = a²(1 + 2cos θ + cos² θ) + a²sin² θ
    = a²(1 + 2cos θ + cos² θ + sin² θ)
    = a²(1 + 2cos θ + 1) = a²(2 + 2cos θ)
    = 2a²(1 + cos θ)
  3. Use identity: 1 + cos θ = 2cos²(θ/2)
    r² + (dr/dθ)² = 2a²(2cos²(θ/2)) = 4a²cos²(θ/2)
  4. √(r² + (dr/dθ)²) = √(4a²cos²(θ/2)) = |2a cos(θ/2)|.
    The full curve is from 0 to 2π. Due to symmetry, we can integrate from 0 to π and multiply by 2. In [0, π], θ/2 is in [0, π/2], so cos(θ/2) is positive.
    s = 2 * ∫ [0 to π] 2a cos(θ/2) dθ
  5. s = 4a * [ 2sin(θ/2) ] from 0 to π
    s = 8a * [ sin(π/2) - sin(0) ] = 8a * [ 1 - 0 ] = 8a

Volumes of Solid of Revolution (About Axes)

This method finds the volume of a 3D solid generated by revolving a 2D plane region around an axis.

Disk Method (Revolution about x-axis)

If the region under y = f(x) from x = a to x = b is revolved around the x-axis:

V = π * ∫ [a to b] [f(x)]² dx = π * ∫ [a to b] y² dx

Disk Method (Revolution about y-axis)

If the region bounded by x = g(y) from y = c to y = d is revolved around the y-axis:

V = π * ∫ [c to d] [g(y)]² dy = π * ∫ [c to d] x² dy

Washer Method (Region between two curves)

Revolving the region between y = f(x) (upper) and y = g(x) (lower) around the x-axis:

V = π * ∫ [a to b] ( [f(x)]² - [g(x)]² ) dx = π * ∫ [a to b] (y_outer² - y_inner²) dx
Common Mistake: Do NOT integrate (y_outer - y_inner) and then square it. You must integrate the difference of the squares.

Surfaces of Solid of Revolution (About Axes)

This finds the surface area of the solid generated by revolving a curve (not a region) around an axis.

Revolution about x-axis

The surface area S generated by revolving the curve y = f(x) from x = a to x = b around the x-axis is:

S = 2π * ∫ [a to b] y * ds = 2π * ∫ [a to b] f(x) * √(1 + (dy/dx)²) dx

Revolution about y-axis

The surface area S generated by revolving the curve y = f(x) from x = a to x = b around the y-axis is:

S = 2π * ∫ [a to b] x * ds = 2π * ∫ [a to b] x * √(1 + (dy/dx)²) dx
Mnemonic: The formula is always S = 2π * ∫ (radius) * (arc length) You then substitute the appropriate formula for 'ds' (Cartesian, polar, or parametric).

Common Curves for Tracing

Being familiar with the shapes of these curves is essential for problems in this unit.