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Chapter 11 - INTEGRATION - Learning objectives
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<h1>Chapter 11 - Integration of MV functions</h1>
<h4>Learning objectives</h4>

<h3>Double integrals</h3>

<ul><li>Evaluating double integrals over rectangular intervals,
$\int_a^b\int_c^d f(x,y)\,dx\,dy$,
in <b>Cartesian coordinates</b>
using partial integration.  In either order (Fubini's theorem).
<li>Evaluating integrals like $$\int_a^b\int_{y=g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx.$$
  <ul>
  <li>Interpreting this integral as the integral over the area $a\lt x \lt b$ and between the curves $g_1(x)\lt y \lt g_2(x)$.
  <li>Order matters!
  <li>Being able to sketch the area of integration,
  <li>Adjusting the limits of integration to change the order of integration.
  </ul>
<li>Double integrals in <b>polar coordinates</b>, like $\iint f(r,\theta)\,dA$.
<ul>
<li>Conversions between $x,y,z \leftrightarrow r,\theta,z$ coordinate systems.
   <li>Being able to set the limits on $r$ and $\theta$ from a sketch of the area of integration.  Or, being able to interpret the limits on $r$ and $\theta$ integrals as an area of integration.
   <li>Using the double integral differential of area: $dA\equiv rdr\,d\theta$ when integrating over an area in 2-d.
  </ul>

<li>Using an integral like $\iint dA=A$ to find the area of... the area of integration.
</ul>

<h3>Triple integrals</h3>

<ul><li>Evaluating triple integrals over rectangular intervals,
$\int_a^b\int_c^d \int_g^h f(x,y,z)\,dx\,dy\,dz$,
in <b>Cartesian coordinates</b>
using partial integration.  In either order (Fubini's theorem).
<li>Evaluating integrals like $$\int_a^b\int_{y=g_1(x)}^{g_2(x)}\int_{z=h_1(x,y)}^{h_2(x,y)} f(x,y,z)\,dz\,dy\,dx.$$
  <ul>
  <li>Interpreting this integral in terms of a volume of integration running from $a\lt x \lt b$,  between the curves $g_1(x)\lt y \lt g_2(x)$, and between the surfaces
  $h_1(x,y)\lt z \lt h_2(x,y)$.
  <li>Order matters!
  <li>Being able to sketch or recognize the volume of integration,
  <li>Adjusting the limits of integration to change the order of integration.
  </ul>
<li>Triple integrals in <b>cylindrical coordinates</b>, like $\iiint f(r,\theta,z)\,dv$.
<ul>
<li>Conversions between $x,y,z \leftrightarrow r,\theta,z$ coordinate systems.
   <li>Being able to set the limits on $r$, $\theta$, and $z$ from a sketch of the volume of integration.  Or, being able to interpret the limits on $r$, $\theta$, and $z$ integrals as an volume of integration.
   <li>Using the triple integral differential of area: $dV\equiv rdr\,d\theta\,dz$ when integrating over a volume in 3-d.
  </ul>
  
<li>Triple integrals in <b>spherical-polar coordinates</b>, like $\iiint f(\rho,\theta,\phi)\,dv$.
<ul>
<li>Conversions between $x,y,z \leftrightarrow r,\theta,\phi$ coordinate systems.
   <li>Being able to set the limits on $\rho$, $\theta$, and $\phi$ from a sketch of the volume of integration.  Or, being able to interpret the limits on $\rho$, $\theta$, and $\phi$ integrals as an volume of integration.
   <li>Using the triple integral differential of area: $dV\equiv \rho^2dr\,d\theta\,\sin\phi d\phi$ when integrating over a volume in 3-d.
  </ul>

<li>Using an integral like $\iiint dV=V$ to find the Volume of... the volume of integration.

</ul>



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