Introduction to Double Integrals & Review of Single-Variable Integration

Course Context & Objectives

• Follows directly after last term’s Introductory Calculus.

  • Extends differentiation and integration ideas to multiple dimensions (mainly 3-D).

  • Goal: develop tools/theorems useful for formulating physical laws in higher-dimensional settings.

• Early lectures include deliberate overlap with previous material to reinforce notation and methods.

Classroom & Logistics Notes

• Lecturer checks audibility and board visibility (“Can you hear/read at the back?”).

• Students encouraged to alert the lecturer if handwriting becomes too small or slanted.

• Working assumption for the entire course: functions and regions will be “well-behaved” (continuous or piece-wise continuous; no pathological edge cases).

Recap — Informal Definition of a Single (Line) Integral

• For a real-valued function $f(x)$ on a closed interval $[a,b]$:

  1. Subdivide $[a,b]$ into $m$ sub-intervals of (possibly equal) width $\Delta x$.

  2. Pick sample points $x<em>1,x</em>2,\dots ,x_m$ inside each sub-interval.

  3. Form the Riemann sum $\displaystyle \sum<em>{i=1}^{m} f(x</em>i)\,\Delta x$.

  4. Define the integral
     $\int<em>a^b f(x)\,dx = \lim</em>{\Delta x \to 0}\,\sum<em>{i=1}^{m} f(x</em>i)\,\Delta x\quad\text{(provided the limit exists)}.$

• Interpretation: area under the curve between $x=a$ and $x=b$.

• Existence of the limit is guaranteed if $f$ is continuous (sufficient but not necessary).

• Using unequal sub-intervals does not change the limit for continuous $f$.

Transition to Two Dimensions

• Aim: generalise the preceding idea to obtain double integrals that compute volumes under surfaces.

• Consider:

  • A planar region $R$ in the $xy$-plane.

  • A scalar field $\psi(x,y)$ defined on that region (picture a surface hovering over $R$).

Informal Definition of the Double Integral

1. Partition $R$ into $n$ small subregions (often rectangles) of area $\Delta A = \Delta x\,\Delta y$.

2. Let $\psi_i$ be the value of $\psi$ at (e.g.) the centre of the $i$-th subregion.

3. Form the sum $\displaystyle \sum<em>{i=1}^{n} \psi</em>i\,\Delta A$.

4. Take the limit as every subregion’s area shrinks:
   $\iint<em>R \psi\,dA = \lim</em>{\Delta A \to 0}\,\sum<em>{i=1}^{n} \psi</em>i\,\Delta A$.

Interpretation: volume between the surface $z = \psi(x,y)$ and the region $R$ in the plane.

Conditions for Existence

• If $\psi$ is piece-wise continuous and $R$ is a suitably “nice” (non-pathological) domain, the limit exists and equals the double integral.

Properties of Double Integrals (Inherited from 1-D Integration)

1. Linearity

• For constants $a,b$ and functions $f,g$:
  $\iint<em>R \big(a\,f(x,y)+b\,g(x,y)\big)\,dA = a\,\iint</em>R f(x,y)\,dA + b\,\iint_R g(x,y)\,dA.$

2. Order (Comparison) Property

• If $f(x,y) \ge g(x,y)$ for every $(x,y) \in R$, then
  $\iint<em>R f(x,y)\,dA \ge \iint</em>R g(x,y)\,dA.$

3. Domain Splitting

• If $R = R<em>1 \cup R</em>2$ with $R<em>1 \cap R</em>2 = \varnothing$, then for any integrable $h$,
  $\iint<em>R h\,dA = \iint</em>{R<em>1} h\,dA + \iint</em>{R_2} h\,dA.$

• Practical impact: choose to split regions whenever it simplifies limits of integration or integrand behaviour.

Computational Strategy Insights

• Choice of integration order ($dx$ first vs. $dy$ first) can drastically affect effort; properties above legitimise re-ordering where permitted.

• Always verify continuity/piece-wise continuity and region simplicity before applying informal limit arguments.

Miscellaneous Transcript Elements (Non-Mathematical)

• Advert-like insertions (“I want to compound my money… Acorns”, “Trade crypto with Robinhood”, “I’m looking for eight ferocious warriors…”).

  • These fragments are unrelated to the calculus content and can be safely ignored for study purposes.

• “Suitable region” phrasing used by lecturer is a catch-all to avoid pathological sets; course deliberately avoids such complications.

These notes capture every mathematical definition, property, example description, and logistical remark made in the transcript while filtering out irrelevant advertorial interruptions.