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
- Sketch the curves! This is the most important step. You need to see which function is on top.
- 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).
- Set up the integral: ∫ [a to b] (y_upper - y_lower) dx.
- 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
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).
- dr/dθ = -a sin θ
- 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) - Use identity: 1 + cos θ = 2cos²(θ/2)
r² + (dr/dθ)² = 2a²(2cos²(θ/2)) = 4a²cos²(θ/2) - √(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 - 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
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
- Revolving around x-axis: The radius of revolution for a point (x, y) is its y-coordinate. So, S = 2π ∫ y ds.
- Revolving around y-axis: The radius of revolution is the x-coordinate. So, S = 2π ∫ x ds.
Common Curves for Tracing
Being familiar with the shapes of these curves is essential for problems in this unit.
- Parabolas: y² = 4ax, x² = 4ay
- Ellipses/Circles: x²/a² + y²/b² = 1
- Hyperbolas: x²/a² - y²/b² = 1
- Semicubical Parabola: ay² = x³ (cusp at origin)
- Cissoid of Diocles: y²(2a - x) = x³ (vertical asymptote at x=2a)
- Strophoid: y² = x² * (a - x) / (a + x) (loop between 0 and a)
- Witch of Agnesi: y = 8a³ / (x² + 4a²)
- Polar: Cardioid: r = a(1 ± cos θ) or r = a(1 ± sin θ)
- Polar: Lemniscate of Bernoulli: r² = a²cos(2θ) (figure-eight shape)
- Polar: Roses: r = a cos(nθ) or r = a sin(nθ) (n-petals if n is odd, 2n-petals if n is even)
- Polar: Spirals: r = aθ (Archimedean), r = eᵃᶿ (Logarithmic)