Answer
$m$ = $\frac{1}{8}(1-3e^{-2})$
$(\frac{1-5e^{-2}}{1-3e^{-2}},\frac{8(1-4e^{-3})}{27(1-3e^{-2})})$
Work Step by Step
$m$ = $\int_0^1\int_0^{e^{-x}}xydydx$
$m$ = $\int_0^1[\frac{xy^2}{2}]_0^{e^{-x}}dx$
$m$ = $\frac{1}{2}\int_0^1xe^{-2x}dx$
$m$ = $\frac{1}{2}[-\frac{xe^{-2x}}{2}-\frac{e^{-2x}}{4}]_0^1$
$m$ = $\frac{1}{8}(1-3e^{-2})$
$x̄$ = $\frac{1}{m}\int_0^1\int_0^{e^{-x}}x^2ydydx$
$x̄$ = $\frac{8}{1-3e^{-2}}\int_0^1\int_0^{e^{-x}}x^2ydydx$
$x̄$ = $\frac{4}{1-3e^{-2}}\int_0^1[x^2y^2]_0^{e^{-x}}dx$
$x̄$ = $\frac{4}{1-3e^{-2}}\int_0^1x^2e^{-2x}dx$
$x̄$ = $\frac{4}{1-3e^{-2}}[-\frac{x^2e^{-2x}}{2}-\frac{xe^{-2x}}{2}-\frac{e^{-2x}}{4}]_0^1$
$x̄$ = $\frac{1-5e^{-2}}{1-3e^{-2}}$
$ȳ$ = $\frac{1}{m}\int_0^1\int_0^{e^{-x}}xy^2dydx$
$ȳ$ = $\frac{8}{1-3e^{-2}}\int_0^1\int_0^{e^{-x}}xy^2dydx$
$ȳ$ = $\frac{8}{3(1-3e^{-2})}\int_0^1[xy^3]_0^{e^{-x}}dx$
$ȳ$ = $\frac{8}{3(1-3e^{-2})}\int_0^1xe^{-3x}dx$
$ȳ$ = $\frac{8}{3(1-3e^{-2})}[-\frac{xe^{-3x}}{3}-\frac{e^{-3x}}{9}]_0^1$
$ȳ$ = $\frac{8(1-4e^{-3})}{27(1-3e^{-2})}$
