H2 Maths Notes (JC 1-2): 5.4) Definite Integrals

Before you revise\ Always sketch the region before integrating. Label intercepts, intersection points, and orientation so you do not mix up top/bottom or left/right functions.

Fundamental Theorem

• If \( F'(x) = f(x) \), then \( \int_a^b f(x) \space dx = F(b) - F(a) \).
• Properties: linearity, additivity over intervals, and reversal of limits introducing a negative sign.

Areas Between Curves

• For vertical slices: \( \int_a^b \bigl( y_\text{top} - y_\text{bottom} \bigr) \space dx \).
• For horizontal slices: \( \int_c^d \bigl( x_\text{right} - x_\text{left} \bigr) \space dy \).
• Determine intersection points by solving \( y_\text{top} = y_\text{bottom} \).

Example -- Area enclosed

Find area between \( y = x^2 \) and \( y = 2x + 3 \).

1. Intersections solve \( x^2 = 2x + 3 \) ⇒ \( x = -1, 3 \).
2. Area \( = \int_{-1}^3 \bigl(2x + 3 - x^2\bigr) dx = \left[ x^2 + 3x - \tfrac{x^3}{3} \right]_{-1}^3 = \tfrac{32}{3} \).

Volumes of Revolution

• About x-axis: \( V = \pi \int_a^b y^2 \space dx \).
• About y-axis: \( V = \pi \int_c^d x^2 \space dy \).
• Shell method (if convenient): \( V = 2\pi \int_a^b x y \space dx \).

Example -- Volume

Region under \( y = \sqrt{x} \) from \( x = 0 \) to \( x = 4 \) revolved about x-axis: \[ V = \pi \int_0^4 x \space dx = \pi \left[ \tfrac{x^2}{2} \right]_0^4 = 8\pi. \]

Mean Value and Average

• Average value of \( f(x) \) on \( [a, b] \): \( \bar{f} = \frac{1}{b - a} \int_a^b f(x) \space dx \).
• For probability density functions, integrals compute probabilities and expected values.

Integration with Modulus or Piecewise Functions

• Identify breakpoints where expression changes sign.
• Split integral into segments where expression simplifies without modulus.

Example -- Modulus

Evaluate \( \int_{-2}^3 \lvert x - 1 \rvert dx \).

1. Break at \( x = 1 \): \ \[ \int_{-2}^1 (1 - x) dx + \int_1^3 (x - 1) dx = \left[ x - \tfrac{x^2}{2} \right]_{-2}^1 + \left[ \tfrac{x^2}{2} - x \right]_1^3 = \tfrac{13}{2}. \]

Improper Definite Integrals

• Use limits for infinite bounds or discontinuities within interval.
• Example: \( \int_0^1 \frac{1}{\sqrt{x}} dx = \lim_{\epsilon \to 0^+} \int_{\epsilon}^1 x^{-1/2} dx = 2 \).

Calculator Workflow

• Use GC definite integral function to verify numeric value after completing manual steps.
• When dealing with piecewise functions, evaluate each segment separately on the GC as a check.
• Document the command (e.g. ∫(2x+3-x^2,x,-1,3)) in working if used.

Exam Watch Points

• Always sketch the region and label axes/limits.
• For revolutions, specify axis clearly and include \( \pi \) factor.
• State units (square or cubic) when context demands.
• Justify convergence when evaluating improper integrals.

Quick Revision Checklist

• [ ] Compute areas between curves accurately with correct limits and integrands.
• [ ] Derive volumes of revolution using shell or disk method as appropriate.
• [ ] Handle modulus and piecewise integrals by splitting at critical points.
• [ ] Evaluate improper integrals with proper limit notation and convergence checks.

Next steps: Finish Section 5.5 on differential equations to complete the calculus module.