Mathematics, 31.01.2020 22:45, ninilizovtskt
Find the area of the region. use a graphing utility to verify your result. integral 0 to 7 9x · cubed root 3x + 1 dx
Answers: 1
![\bf \displaystyle \int\limits_{0}^{7}\ 9x\sqrt[3]{3x+1}\cdot dx\\\\ -----------------------------\\\\ u=3x+1\implies \cfrac{du}{dx}=3\implies \cfrac{du}{3}=dx\\\\ -----------------------------\\\\ \displaystyle \int\limits_{0}^{7}\ 9x\sqrt[3]{u}\cdot \cfrac{du}{3}\implies \int_{0}^{7}\ 3x\sqrt[3]{u}\cdot dx\\\\ -----------------------------\\\\ now\qquad u=3x+1\implies u-1=3x\\\\ -----------------------------\\\\](/tpl/images/0491/4032/9a1e6.png)
![\bf \displaystyle \int\limits_{1}^{22}\ \left( u^{\frac{4}{3}}-u^{\frac{1}{3}} \right) dx\implies \cfrac{u^{\frac{7}{3}}}{\frac{7}{3}}-\cfrac{u^{\frac{4}{3}}}{\frac{4}{3}}\implies \left. \cfrac{3\sqrt[3]{u^7}}{7}-\cfrac{3\sqrt[3]{u^4}}{4}\right]_{1}^{22}](/tpl/images/0491/4032/45ac0.png)
![\bf \left[ \cfrac{3\cdot 1356.187}{7} \right]-\left[ \cfrac{3\cdot 61.645}{4} \right] \\\\\\ 581.22-46.234\approx 534.98936648870](/tpl/images/0491/4032/09173.png)
![\bf \left[ \cfrac{3}{7} \right]-\left[ \cfrac{3}{4} \right]\implies -\cfrac{9}{28} \\\\\\ thus \\\\\\ 534.98936648870355525281-\left( -\cfrac{9}{28} \right) \\\\\\ 534.98936648870355525281+\cfrac{9}{28} \approx 535.31079506013212668138](/tpl/images/0491/4032/fda79.png)
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Find the area of the region. use a graphing utility to verify your result. integral 0 to 7 9x · cube...
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