Chapter 5, Problem 416RE

True or False? Justify your answer with a proof or a counterexample.

416. ${\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)dydx}={\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)dydx}}}$

To determine

To calculate:

True or False? Justify your answer with a proof or a counterexample.

Answer to Problem 416RE

The statement ${\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}={\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dydx}$ is true.

Explanation of Solution

Given Information: ${\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}={\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dydx}$

Calculation:

Since $f\left(x,y\right)$ is integral function

Suppose that $f\left(x,y\right)$ is differentiable and continuous function

Let $f\left(x,y\right)=g\text{'}\left(x\right)h\text{'}\left(y\right)$

Now consider the LHS integration

${\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}={\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}g\text{'}\left(x\right)h\text{'}\left(y\right)}dydx}$

Now integrate with respect to $dy$.

$\begin{array}{l}{\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}={\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}g\text{'}\left(x\right)h\text{'}\left(y\right)}dydx}\\ ={\displaystyle \underset{a}{\overset{b}{\int }}g\text{'}\left(x\right){\left(h\left(y\right)\right)}_c^{d}dx}\end{array}$

Now put the limits of y and minus lower limit form upper limit

This implies

$\begin{array}{l}{\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}={\displaystyle \underset{a}{\overset{b}{\int }}g\text{'}\left(x\right){\left(h\left(y\right)\right)}_c^{d}dx}\\ ={\displaystyle \underset{a}{\overset{b}{\int }}g\text{'}\left(x\right)\left(h\left(d\right)-h\left(c\right)\right)dx}\\ =\left[h\left(d\right)-h\left(c\right)\right]{\displaystyle \underset{a}{\overset{b}{\int }}g\text{'}\left(x\right)dx}\end{array}$

Now integrate with respect to x

This implies

$\begin{array}{l}{\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}=\left[h\left(d\right)-h\left(c\right)\right]{\displaystyle \underset{a}{\overset{b}{\int }}g\text{'}\left(x\right)dx}\\ =\left[h\left(d\right)-h\left(c\right)\right]{\left(g\left(x\right)\right)}_a^b\end{array}$

Now put the limits of x and minus lower limit form upper limit

This implies

$\begin{array}{l}{\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}=\left[h\left(d\right)-h\left(c\right)\right]{\left(g\left(x\right)\right)}_a^b\\ =\left[h\left(d\right)-h\left(c\right)\right]\left(g\left(b\right)-g\left(a\right)\right)\\ =\left[g\left(b\right)-g\left(a\right)\right]\left[h\left(d\right)-h\left(c\right)\right]\\ {\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}=\left[g\left(b\right)-g\left(a\right)\right]\left[h\left(d\right)-h\left(c\right)\right]\end{array}$

Now consider the RHS integration

${\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dydx}={\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}g\text{'}\left(x\right)h\text{'}\left(y\right)}dydx}$

Now integrate with respect to x

This implies

$\begin{array}{l}{\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dxdy}={\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}g\text{'}\left(x\right)h\text{'}\left(y\right)dxdy}}\\ ={\displaystyle \underset{c}{\overset{d}{\int }}h\text{'}\left(y\right){\left(g\left(x\right)\right)}_a^{b}dy}\end{array}$

Now put the limits of x and minus lower limit form upper limit

This implies

$\begin{array}{l}{\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dxdy}={\displaystyle \underset{c}{\overset{d}{\int }}h\text{'}\left(y\right){\left(g\left(x\right)\right)}_a^{b}dy}\\ ={\displaystyle \underset{c}{\overset{d}{\int }}h\text{'}\left(y\right)\left(g\left(b\right)-g\left(a\right)\right)dy}\\ =\left[g\left(b\right)-g\left(a\right)\right]{\displaystyle \underset{c}{\overset{d}{\int }}h\text{'}\left(y\right)dy}\end{array}$

Now integrate with respect to $dy$

$\begin{array}{l}{\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dxdy}=\left[g\left(b\right)-g\left(a\right)\right]{\displaystyle \underset{c}{\overset{d}{\int }}h\text{'}\left(y\right)dy}\\ =\left[g\left(b\right)-g\left(a\right)\right]{\left(h\left(y\right)\right)}_c^{d}dy\end{array}$

Now put the limits of x and minus lower limit form upper limit

This implies

$\begin{array}{l}{\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dxdy}=\left[g\left(b\right)-g\left(a\right)\right]{\left(h\left(y\right)\right)}_c^d\\ {\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dxdy}=\left[g\left(b\right)-g\left(a\right)\right]\left[h\left(d\right)-h\left(c\right)\right]\end{array}$

Hence, the statement ${\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}f\left(x,y\right)}dydx}={\displaystyle \underset{c}{\overset{d}{\int }}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x,y\right)}dydx}$ is true.