Optimizing Integration: Sketching vs. Graphing for Type I, II, and III Problems

[mathrocks]

If you want to find volume or area using double/triple integration, is graphing the (r) and (d) the only way to see if it is a type I, II, or III problem? I'm not really good at sketching planes so I'm having great difficulty finding the answer. But when it is already setup I can do it easily.

 

[fourier jr]

What are type I, II, III? never heard of that before.

 

[mathrocks]

What are type I, II, III? never heard of that before.

They are just names for how we setup the limits of integration. A type one for double integration is when D={(x,y)| a<=x<=b, g1(x)<=y<=g2(x)}

Type II is D={(x,y)| c <= x <= d, h1(y) <= x <= h2(y)}

 

[jatin9_99]

that depend upon how well u set up the problem, but skecthing helps

 

[speeding electron]

Couldn't you choose? For example, for a double integral, where the region to integrate over is the triangle with vertices (0,0), (2,0), (0,1), the region could be set with:

D = {x,y : 0 <= y <= -x/2 + 1 , 0 <= x <= 2} (type 1)

or:

D = {x,y : 0 <= x = 2y + 2 , 0 <= y <= 1} (type 2).

But then of course to see the limiting function for either of these, sketching helps.