# Find: Explain the statement: If for a given function f[x,y] and a given region R in the...

## Question:

Explain the statement: If for a given function {eq}f[x,y] {/eq} and a given region R in the xy-plane, it turns out that {eq}\displaystyle \int \int_R f[x,y] dxdy <0 {/eq} then you are guaranteed that there are some points {eq}\left \{ x,y \right \}\: in\: R \:with\: f[x,y] < 0. {/eq}?

## Riemann Sums for Double Integrals:

We model Riemann Sums for double integrals after those for a single integral. Suppose {eq}f(x,y) {/eq} is defined over {eq}R {/eq} (a region in the {eq}xy {/eq}-plane). The double integral {eq}\iint_D f(x,y)\, dA {/eq} can then be approximated by breaking {eq}R {/eq} into sub-rectangles and approximating the signed volume above or below the rectangle as {eq}\sum f(P_i)\Delta A_i {/eq} where {eq}P_i {/eq} is a sample point in the {eq}i {/eq}th sub-rectangles and {eq}\Delta A_i {/eq} is the area of the {eq}i {/eq}th sub-rectangle.

The double integral {eq}\iint_D f(x,y)\, dA {/eq} is equal to the limit of {eq}\sum f(P_i)\Delta A_i {/eq} as we let the maximum area of a sub-rectangle go to 0.

The contrapositive of the statement that we wish to explain is

If the set {eq}\{ (x,y)\in R\mid f(x,y)<0\} {/eq} is the empty set, then \iint_R f(x,y)\, dx\, dy\geq 0.

If we suppose that {eq}\{(x,y)\in R\mid f(x,y)<0\} {/eq} is empty. This tells us that {eq}f(x,y)\geq 0 {/eq} for all {eq}(x,y)\in R {/eq}. It follows that every Riemann Sum {eq}\sum f(P_i)\Delta A_i\geq 0 {/eq} since {eq}\Delta A_i {/eq} is an area and {eq}f(x,y)\geq 0 {/eq} for all {eq}(x,y)\in R {/eq}.

Therefore, the statement is true. 