Double Integral

Example from class: Fubini's Theorem Rectangular Region

Integrate[Integrate[x*Sin[x*y],{x,0,Pi/2}],{y,0,1}]

Changing the order of integration:

Integrate[Integrate[x*Sin[x*y],{y,0,1}],{x,0,Pi/2}]

Example from class: General Region

Integrate[Integrate[((4y)/((x^3)+2)),{y,0,2x}],{x,1,2}]

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Changing the order of integration, you can see that this integral would be a whopper to evaluate by hand!

Integrate[Integrate[((4y)/((x^3)+2)),{x,1,2}],{y,0,2}]+Integrate[Integrate[((4y)/((x^3)+2)),{x,y/2,2}],{y,2,4}]

Simplify[%]

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Plotting Regions

You can use Mathematica to plot the region of integration, which might be helpful as you solve problems.

For the region R={(x,y)| 1≤x≤2, 0≤y≤2x}:

RegionPlot[1≤x≤2&&0≤y≤2x,{x,0,5},{y,0,5},PlotPoints->30]

For the region R={(x,y)| 3y≤x≤3, 0≤y≤1}:

RegionPlot[3y≤x≤3&&0≤y≤1,{x,0,5},{y,0,5},PlotPoints->30]

For the region R={(x,y)| 0≤x≤3, 0≤y≤x/3}:

RegionPlot[0≤x≤3&&0≤y≤x/3,{x,0,5},{y,0,5},PlotPoints->30]

For the region R={(x,y)| 1-y≤x≤(Sqrt[1-(y^2)]), 0≤y≤1}:

RegionPlot[1-y≤x≤(Sqrt[1-(y^2)])&&0≤y≤1,{x,0,2},{y,0,2},PlotPoints->30]

For the region R={(x,y)| 0≤x≤2, (x^2)≤y≤2x}:

RegionPlot[0≤x≤2&&(x^2)≤y≤2x,{x,0,2},{y,0,4},PlotPoints->30]

You should be able to do the ones above by hand, but Mathematica can help you if the region is very complicated (this is a region you would have to split up into multiple regions to be able to integrate over:

RegionPlot[0≤x≤2&&(x^2)≤y≤2x &&1≤x*y≤6,{x,0,2},{y,0,4},PlotPoints->30]