Question

Double Integral True or False

Answer 1

1.\[\int\limits_{}^{}\int\limits_{R}^{} f(x,y) dA>0\] implies that f lies entirely above the xy-plane.

2.If a<b and c<d, then \[\int\limits_{a}^{b}\int\limits_{c}^{d}(1+x^2)e ^{1-y^2} dydx>0.\]

3.If a<b and c<d then the average value of z=ex−y on the rectangle R=[a,b]×[c,d] is \[\frac{ 1 }{ ((b−a)(d−c)) } \int\limits_{c}^{d}\int\limits_{a}^{b} e^(x−y) dxdy.\]

4.If we attempt to optimize z=2x^2+2y^2 subject to x^2+y^2=5 we find that the maximum value is 10 and the minimum value is also 10 and they occur an infinite number of times.

Answer 2

@Kainui

Answer 3

True

Answer 4

true @zaner336

Answer 5

@Tyler.Evan2 & @zaner336 just writing "true" doesn't help me

Answer 6

Imagine f(x,y) as this sheet of paper going both above and below the x-y plane. Let's look at a cross section of it:

Is the integral positive or negative now? Well it's ambiguous, maybe the positive part is more than the negative contribution so the overall integral is positive. But wait, that means having an integral greater than 0 doesn't mean that the surface is entirely above the x-y plane, it just means most of it is above the xy plane which could be all of it or just some of it!