Chapter 16.1, Problem 56E

Product of integrals Suppose f(x, y) = g(x)h(y), where g and h are continuous functions for all real values of x: and y.

a.    Show that ${\displaystyle {\int }_c^d{\displaystyle {\int }_a^{b}f\left(x,y\right)dxdy=\left({\displaystyle {\int }_a^{b}g\left(x\right)dx}\right)\left({\displaystyle {\int }_c^{d}h\left(y\right)dy}\right)}}$ Interpret this result geometrically.

b.    Write ${\left({\displaystyle {\int }_a^{b}g\left(x\right)dx}\right)}^2$ as an iterated integral.

c.    Use the result of part (a) to evaluate ${\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{10}^{30}{e}^{-4y^2}\cos xdy}dx.}$