Chapter 16.1, Problem 51E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a.    The region of integration for ${\displaystyle {\int }_4^6{\displaystyle {\int }_1^{3}4dxdy}}$ is a square.

b.    If f is continuous on ${ℝ}^2$, then ${\displaystyle {\int }_4^6{\displaystyle {\int }_1^{3}f\left(x,y\right)dxdy=}}{\displaystyle {\int }_4^6{\displaystyle {\int }_1^{3}f\left(x,y\right)dydx}}$.

c.    If f is continuous on ${ℝ}^2$, then ${\displaystyle {\int }_4^6{\displaystyle {\int }_1^{3}f\left(x,y\right)dxdy=}}{\displaystyle {\int }_1^3{\displaystyle {\int }_4^{6}f\left(x,y\right)dydx}}$.