MATH1011 - Lecture 14

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MATH1011 - Lecture 13
Skeleton Slides

Double Integrals

• We can use double integrals to find the volume of a surface
  • i.e. volume underneath a surface formed by $z=f(x,y)$
• Region $R=\{(x,y)|a\le x\le b\text{and}c\le y\le d\}=[a,b]\times [c,d]$
  
• We are given the double integral: (see slides) $\underset{R}{\iint }f(x,y)dA={\int }_a^b\left[{\int }_c^{d}f(x,y)dy\right]dx$
  • If $f(x,y)\ge 0$ for all $(x,y)$ in $R$, then the double integral is the volume of the solid bounded by $z=f(x,y)$ and $xy$-plane over $R$
    • $dA$ for this example would equal $dxdy$
  • Order does not matter as long as the bounds match the variable

Regions Bounded By Functions: Type I

• Let a given region $R$ be bounded by the functions $g(x),h(x)$
  • $g(x)\le h(x)$ for all $x\in [a,b]$
  • The shape is enclosed by $a\le x\le b$ and $g(x)\le y\le h(x)$
  • i.e. $x$ is bounded by constants and $y$ is bounded by functions of $x$
• The volume below $f(x,y)$ over $R$ is given by the double integral: $\underset{R}{\iint }f(x,y)dA={\int }_a^b\left[{\int }_{g(x)}^{h(x)}f(x,y)dy\right]dx$

Regions Bounded By Functions: Type II

• Let a given region $R$ be bounded by the functions $G(y),H(y)$
  • $G(y)\le H(y)$ for all $y\in [a,b]$
  • The shape is enclosed by $G(y)\le x\le H(y)$ and $c\le y\le d$
  • i.e. $y$ is bounded by constants and $x$ is bounded by functions of $y$
• The volume below $f(x,y)$ over $R$ is given by the double integral: $\underset{R}{\iint }f(x,y)dA={\int }_c^d\left[{\int }_{G(y)}^{H(y)}f(x,y)dx\right]dy$

For regions bounded by functions, if you are trying to find area in a 2D space, set $f(x,y)$ to $1$. Example of area of eclipse on lecture recording.

Fubini’s Theorem

$I={\int }_a^b\left[{\int }_{g(x)}^{h(x)}f(x,y)dy\right]dx={\int }_c^d\left[{\int }_{G(y)}^{H(y)}f(x,y)dx\right]dy$

More Complicated Regions

• When the region $R$ is neither type I or type II, we can usually describe is as a union of several sub-regions
  • $R=R_1\cup R_2\cup \cdots \cup R_k$
• Thus: $\underset{R}{\iint }f(x,y)dA=\underset{R_1}{\iint }f(x,y)dA+\underset{R_2}{\iint }f(x,y)dA+\cdots +\underset{R_k}{\iint }f(x,y)dA$

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We will miss next Friday’s lesson, thus we are starting content early

Triple Integrals

• The triple integral of $f$ over $R$ is: $\underset{R}{\iiint }f(x,y,z)dV=\underset{R}{\iiint }f(x,y,z)dxdydz$

Fubini’s Theorem Revisited

$I={\int }_a^b\left[{\int }_c^d\left({\int }_p^{q}f(x,y,z)dz\right)dy\right]dx$

This equation is equal to all forms of its rearrangement

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MATH1011 - Lecture 15
Annotated Slides