15.2 Videos Guide

15.2a

• The double integral over general regions

  • $\iint^{\ }_{D}{f(x,\ y)\ dA}$
    $=\int^{b}_{a}{\int^{g_{2}(x)}_{g_{1}(x)}{f(x,\ y)\ dydx}}$ if $D=\left\{ (x,\ y)\middle|a\leq x\leq b,\ g_{1}(x)\leq y\leq g_{2}(x)\right\}$

OR $=\int^{d}_{c}{\int^{h_{2}(y)}_{h_{1}(y)}{f(x,\ y)\ dxdy}}$ if $D=\{(x,\ y)|h_{1}(y)\leq x\leq h_{2}(y),\ c\leq y\leq d\}$

Exercises:

15.2b

• Evaluate the iterated integral.

  • $\int^{2}_{0}{\int^{y^{2}}_{0}{x^{2}y\ dx\ dy}}$

15.2c

• $\int^{\frac{\pi}{2}}_{0}{\int^{x}_{0}{x\sin y\ dy\ dx}}$

• Evaluate the double integral.
  $\iint^{\ }_{D}{\left(x^{2}+2y\right)\ dA}$, $D$ is bounded by $y=x$, $y=x^{3}$, $x\geq0$

15.2d

• Find the volume of the solid enclosed by the paraboloid $z=x^{2}+y^{2}+1$ and the planes $x=0$, $y=0$, $z=0$, and $x+y=2$.

• Sketch the region of integration and change the order of integration.
  $$\int^{2}_{0}{\int^{4}_{x^{2}}{f(x,\ y)\ dy\ dx}}$$

15.2e

• Evaluate the integral by reversing the order of integration.
  $$\int^{1}_{0}{\int^{1}_{x^{2}}{\sqrt{y}\sin y\ dy\ dx}}$$