15.1 Videos Guide

15.1a

• Volume using double integrals

  • $V=\iint^ {}_{R}{f(x,\ y)\ dA}$, where $f(x,\ y)\geq0$ over $R$, a rectangular region in $\mathbb{R}^{2}$

Exercise:

Calculate the iterated integral.

• $\int^{1}_{0}{\int^{1}_{0}{(x+y)^{2}\ dx\ dy}}$

15.1b

Theorem (statement):

• Fubini’s Theorem: If $f$ is continuous or has a finite number of discontinuities on $R=\{a\leq x\leq b,\ c\leq y\leq d\}$, then
  $$\iint^ {}_{R}{f(x,\ y)\ dA}=\int^{d}_{c}{\int^{b}_{a}{f(x,\ y)\ dx\ dy}}=\int^{b}_{a}{\int^{d}_{c}{f(x,\ y)\ dy\ dx}}$$

• If $f(x,\ y)=g(x)h(y)$, then $\int^{d}_{c}{\int^{b}_{a}{f(x,\ y)\ dA}}=\int^{b}_{a}{g(x)\ dx}\int^{d}_{c}{h(y)\ dy}$

Exercises:

15.1c

• Estimate the volume of the solid that lies below the surface $z=1+x^{2}+3y$ and above the rectangle $R=\lbrack1,\ 2\rbrack\times\lbrack0,\ 3\rbrack$. Use a Riemann sum with $m=n=2$ and choose the sample points to be lower left corners.
  (b) Use the midpoint rule to estimate the volume in part (a)

• Evaluate the double integral by first identifying it as the volume of a solid.
  $\iint^ {}_{R}{(2x+1)\ dA}$, $R=\left\{ (x,\ y)\middle|0\leq x\leq2,\ 0\leq y\leq4\right\}$

15.1d

• Calculate the iterated integral.
  $$\int^{3}_{1}{\int^{5}_{1}{\frac{\ln y}{xy}\ dy\ dx}}$$

• Calculate the double integral.
  $\iint^ {}_{R}{\left(y+xy^{-2}\right)\ dA}$, $R=\{(x,\ y)|0\leq x\leq2,\ 1\leq y\leq2\}$