Calculate the iterated integral ${\int}_{0}^{2}{\int}_{0}^{3}{e}^{x-y}dydx$

Damien Horton
2022-07-27
Answered

Calculate the iterated integral ${\int}_{0}^{2}{\int}_{0}^{3}{e}^{x-y}dydx$

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i4epdp

Answered 2022-07-28
Author has **12** answers

${\int}_{0}^{2}{\int}_{0}^{3}{e}^{x-y}dydx$

$={\int}_{0}^{2}[-{e}^{x-y}{]}_{0}^{y=3}dx$

$=-{\int}_{0}^{2}(-{e}^{x-3}-{e}^{x})dx$

$=-[{e}^{x-3}-{e}^{x}{]}_{0}^{2}$

$=-[({e}^{-1}-{e}^{2})-({e}^{-3}-{e}^{0})]$

$={e}^{2}-\frac{1}{e}+\frac{1}{{e}^{3}}-1$

$={\int}_{0}^{2}[-{e}^{x-y}{]}_{0}^{y=3}dx$

$=-{\int}_{0}^{2}(-{e}^{x-3}-{e}^{x})dx$

$=-[{e}^{x-3}-{e}^{x}{]}_{0}^{2}$

$=-[({e}^{-1}-{e}^{2})-({e}^{-3}-{e}^{0})]$

$={e}^{2}-\frac{1}{e}+\frac{1}{{e}^{3}}-1$

asked 2022-04-30

State an antiderivative $F:D\u27f6\mathbb{C}$ or explain why an antiderivative does not exist.

State an antiderivative $F:D\u27f6\mathbb{C}$ or explain why an antiderivative does not exist.

a) $f(z)={z}^{2},D=\mathbb{C}$

b) $f(z)=z\mathrm{sin}z,D=\mathbb{C}$

c) $f(z)=|z{|}^{2},D=\mathbb{C}$

d) $f(z)=\overline{z},D=\mathbb{C}$

For a) and b) I just performed simple integration for the equations

a) ${z}^{3}/3+C$ and b) $-z\mathrm{cos}z+\mathrm{sin}z+C$ but I am stuck on c) and d). It seems like an antiderivative does not exist but how do i explain that?

State an antiderivative $F:D\u27f6\mathbb{C}$ or explain why an antiderivative does not exist.

a) $f(z)={z}^{2},D=\mathbb{C}$

b) $f(z)=z\mathrm{sin}z,D=\mathbb{C}$

c) $f(z)=|z{|}^{2},D=\mathbb{C}$

d) $f(z)=\overline{z},D=\mathbb{C}$

For a) and b) I just performed simple integration for the equations

a) ${z}^{3}/3+C$ and b) $-z\mathrm{cos}z+\mathrm{sin}z+C$ but I am stuck on c) and d). It seems like an antiderivative does not exist but how do i explain that?

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