Question

a massa da placa de densidade ro, dada por ro(xy) = yx+1, limitada pelas retas x=0, y=x, y=4 é?

A) 30kg

B) 20kg

C) 40kg

D) 60kg

E) 50kg

Answer 1

Olá

Alternativa correta, letra C) 40 kg

$\mathsf{\delta (x,y)=yx+1}\\\\\\\mathsf{x=0\qquad\qquad y=x\qquad\qquad y=4}$

Observando o gráfico, temos que os limites da primeira integral vai de 0 a 4. e a da segunda integral vai de 'x' à 4; Integrando em dydx


$\displaystyle \mathsf{ \int\limits^4_0 \int\limits^4_x( {yx+1}) \,dy dx }\\\\\\\\\mathsf{ \int\limits^4_0 \left[ \int\limits^4_x( {yx+1}) \,dy \right]dx}\\\\\\\\\mathsf{ \int\limits^4_0 \left[ \frac{y^2}{2}x~+ ~y\bigg|^4_x~ \right]dx}\\\\\\\\\mathsf{ \int\limits^4_0 \left[ \left( \frac{4^2}{2}x~+~4 \right)~-~ \left( \frac{x^2}{2}x~+~x \right)\right]dx}\\\\\\\\\mathsf{ \int\limits^4_0 \left( 7x- \frac{x^3}{2}+4\right) dx}\\\\\\\\\mathsf{\left( \frac{7x^2}{2}- \frac{x^4}{8} +4x\right)\bigg |^4_0}$

$\displaystyle \mathsf{\left( \frac{7\cdot 4^2}{2}- \frac{4^4}{8} +4\cdot 4\right)~- ~\left( \frac{7\cdot 0^2}{2}- \frac{0^4}{8} +4\cdot 0\right)}\\\\\\\\\mathsf{\left(56-32+16\right)~-~0}\\\\\\\\\mathsf{\boxed{40}\qquad\qquad\qquad\qquad \Longrightarrow \qquad\qquad Letra~ C)}$

Answer 2

Resposta:

C) 40kg

Explicação passo a passo: