Q. 13 - Calculus

Question

State Fubini's theorem.

Step-by-Step Solution

Answer

Ans: Then Fubini's theorem states as ${\iint }_{R}f(x,y)dA={\int }_c^d{\int }_a^{b}f(x,y)dxdy={\int }_a^b{\int }_c^{d}f(x,y)dydx$

Step 1. Given information:

Fubini's theorem. 

Step 2. Stating Fubini's theorem:

Fubini's theorem states that a double integral can be computed using iterated integrals, which means the order of integration can be switched for which the order chosen is easier to evaluate. Let $a<b$ and $c<d$ be real numbers, let R be the rectangle defined by $R=\left\{\right(x,y)\mid a\le x\le b$and $c\le y\le d\}$ and let $f(x, y)$ be continuous on R. Then Fubini's theorem states as

${\iint }_{R}f(x,y)dA={\int }_c^d{\int }_a^{b}f(x,y)dxdy={\int }_a^b{\int }_c^{d}f(x,y)dydx$