Question

me ajuda a calcular a area dessa curva pela integral, nessas condições , tem que dar 52/3 , mas nao estou conseguindo

Answer 1

$\displaystyle \sf \int\limits^8_0 \sqrt{x+1}dx \\\\\\ \text{Fa{\c c}amos} : \\\\ u = x+1 \to du = dx \\\\ Da{\'i}}: \\\\ \int\limits^8_0 \sqrt{u}\ du \to \int\limits^8_0 u^{\frac{1}{2}}\ du \\\\\\ \left \frac{u^{\left(\frac{1}{2}+1\right)}}{\frac{1}{2}+1}\right |\limits^8_0 \to \left \frac{u^{\frac{3}{2}}}{\displaystyle \frac{3}{2}} \right |\limits^8_0 \\\\\\ \left \frac{2\cdot \sqrt{(x+1)^3 }}{3} \right |\limits^8_0 \to \frac{2\cdot \sqrt{(8+1)^3}}{3}-\frac{2\cdot \sqrt{(0+1)^3}}{3}$

$\displaystyle \sf \frac{2\cdot \sqrt{9^3}-2}{3}\to \frac{2\cdot \sqrt{3^6}}{3}-\frac{2}{3} \\\\\\ \frac{2\cdot 3^3-2}{3} \to \frac{2\cdot 27-2}{3} \to \frac{54-2}{3} \\\\\\ \huge\boxed{\frac{52}{3}} \checkmark$