Question

O valor da Integral \int _1^3\int _0^{y^1}\int _{-1}^1dvdy

a. -5
b. 1
c. 5
d. 2
e. 0

Answer 1

Boa tarde!

Solução!

$\displaystyle\int_{1}^{3}\displaystyle\int_{0}^{1} \displaystyle\int _{-1}^{1}ydv\\\\\\\ \displaystyle\int_{1}^{3}\displaystyle\int_{0}^{1} \displaystyle\int _{-1}^{1}ydzdydx\\\\\\\ \displaystyle\int_{1}^{3}\displaystyle\int_{0}^{1}ydxdy\bigg[z\bigg]_{-1} ^{1}\\\\\\\\\ \displaystyle\int_{1}^{3}\displaystyle\int_{0}^{1}ydxdy\bigg[1-(-1)\bigg]\\\\\\\ \displaystyle\int_{1}^{3}\displaystyle\int_{0}^{1}ydxdy\bigg[2\bigg]\\\\\\\ \displaystyle\int_{1}^{3}\displaystyle\int_{0}^{1}2ydxdy$


$\displaystyle\int_{1}^{3}dx\bigg[2y\bigg]_{0} ^{1} \\\\\\\ 2\displaystyle\int_{1}^{3}dx\bigg[ \frac{y^{2} }{2} \bigg]_{0} ^{1} \\\\\\\ 2. \dfrac{1}{2}~~ \displaystyle\int_{1}^{3}dx\bigg[ y^{2} \bigg]_{0} ^{1} \\\\\\\ 2. \dfrac{1}{2}~~ \displaystyle\int_{1}^{3}dx\bigg[ 1^{2} -0^{2} \bigg] \\\\\\\ 2. \dfrac{1}{2}~~ \displaystyle\int_{1}^{3}1.dx\\\\\\\ 2. \dfrac{1}{2}~~ \bigg[x\bigg]_{1}^{3}\\\\\\\ 2. \dfrac{1}{2}~~ \bigg[3-1\bigg]\\\\\\\ 2. \dfrac{1}{2}~~ \bigg[2\bigg]$


$\dfrac{2.1.2}{2}= \dfrac{4}{2}=2$


$\boxed{Resposta:~~ \displaystyle\int_{1}^{3}\displaystyle\int_{0}^{1} \displaystyle\int _{-1}^{1}ydv=2~~\boxed{Alternativa~~D}}$

Boa tarde!
Bons estudos!

Answer 2

Resposta:

A alternativa correta é a: D.