Applications of Integration — NDA Maths

Formulas (7)

• The Definite Integral as Signed Area · Area under a curve above the axis
  $$A = \int_a^b f(x)\,dx \quad (f \ge 0)$$
• Area Under a Curve Between Two Lines · Semicircle area shortcut
  $$y = \sqrt{r^2 - x^2}\ \Rightarrow\ A = \tfrac{1}{2}\pi r^2$$
• Below the Axis, Loops & the Factor of 2 · Area with a sign change at c
  $$A = \left|\int_a^c f\,dx\right| + \left|\int_c^b f\,dx\right|$$
• Regions Bounded by Lines & Modulus · Polygon area, not an integral
  $$\text{rectangle } = \text{w}\times\text{h} \qquad \text{triangle } = \tfrac{1}{2}\,b\,h$$
• Area of a Parabola Cut by Its Latus Rectum · Parabola–latus rectum area
  $$y^2 = 4ax:\quad A = 2\int_0^{a}\sqrt{4ax}\,dx = \tfrac{8}{3}a^2$$
• Area Under a Step (Greatest-Integer) Curve · One step = one rectangle
  $$A = |n| \times (\text{interval width}), \quad [x] = n$$
• Area of a Circular Segment by a Chord · Segments of a circle
  $$A_2 = \text{sector} - \text{triangle}, \qquad A_1 = \pi r^2 - A_2$$

Watch out for (7)

• Integrate only where the curve stays above the axis
  → Area Under a Curve Between Two Lines
• The raw integral can be zero while the area is not
  → Below the Axis, Loops & the Factor of 2
• A negative area answer means a missing modulus
  → Below the Axis, Loops & the Factor of 2
• |x| ≤ p gives a side of length 2p, not p
  → Regions Bounded by Lines & Modulus
• Double the half-region, and use the right limit
  → Area of a Parabola Cut by Its Latus Rectum
• Negative step values still give positive area
  → Area Under a Step (Greatest-Integer) Curve
• Subtract the triangle from the sector
  → Area of a Circular Segment by a Chord

Formulas (4)

• Finding Where Two Curves Meet · Intersection condition
  $$f(x) = g(x) \ \Rightarrow\ \text{the } x\text{-values where the curves cross}$$
• Area Between Curves: Top Minus Bottom · Area between curves
  $$A = \int_a^b \bigl(\text{top} - \text{bottom}\bigr)\,dx$$
• Area Between a Curve and a Line · Curve over a line
  $$A = \int_a^b \bigl(y_{\text{curve}} - y_{\text{line}}\bigr)\,dx$$
• Composite Regions: Subtract Areas · Quarter-circle minus a curve
  $$A = \tfrac{1}{4}\pi r^2 - \int_a^b f(x)\,dx$$

Watch out for (5)

• A modulus can create extra intersections
  → Finding Where Two Curves Meet
• The intersection x-values are the limits — don't guess them
  → Finding Where Two Curves Meet
• Subtract top minus bottom, not in equation order
  → Area Between Curves: Top Minus Bottom
• Pick the correct branch of a sideways parabola
  → Area Between a Curve and a Line
• Subtract the area under the curve, not the curve's value
  → Composite Regions: Subtract Areas