Question

For a double integral with 2bounds, how would I convert the bounds to only 1bound? D1:0<=x<=1, -2x+3<=y<=3 and D2: 1<=x<=5, 1/2*x+1/2<=y<3.....How do I combine D1UD2?

Answer 1

Okay, so looking at this, I would say that because your first bounds in dx are 0 to 1, and the second one is 1 to 5, the combined new bund has to be from 0 to 5. That way everything is inclusive for f(x,y) in dx. for dy both limit at 3, so the upper bound is already taken care of. So by subbing in zero for x in the lower bounds, it is seen that (1/2)x+1 gives the smaller value and would include the values for 2x+3 if it were used for the lower bound in dy. Also you can look at the y intercepts and slope of each line and intuit the same conclusion. so your new bounds are\[\int\limits_{0}^{5}\int\limits_{x/2 +1/2}^{3}f(x,y)dydx\]