region enclosed by parametric eqn: x = t^3 - 8t, y = 7t^2
- Thread starter thebenji
- Start date
Try graphing your parametrics.
Solve \(\displaystyle y=7t^{2}\) for t and sub into x
\(\displaystyle t=\sqrt{\frac{y}{7}}\)
\(\displaystyle x=8(\sqrt{\frac{y}{7}})^{3}-8(\sqrt{\frac{y}{7}})\)
\(\displaystyle \L\\=\frac{8(7y)^{\frac{3}{2}}}{343}-\frac{8\sqrt{7y}}{7}\)
Setting this to 0 and solving for y gives y=0 and 56
Solve \(\displaystyle y=7t^{2}\) for t and sub into x
\(\displaystyle t=\sqrt{\frac{y}{7}}\)
\(\displaystyle x=8(\sqrt{\frac{y}{7}})^{3}-8(\sqrt{\frac{y}{7}})\)
\(\displaystyle \L\\=\frac{8(7y)^{\frac{3}{2}}}{343}-\frac{8\sqrt{7y}}{7}\)
Setting this to 0 and solving for y gives y=0 and 56
Re: region enclosed by parametric eqn: x = t^3 - 8t, y = 7t^
Hello, thebenji!
Eliminate the parameter (as Galactus did):
From [1], we have: \(\displaystyle \L\,t \:=\:\sqrt{\frac{y}{7}}\)
Substitute into [2]: \(\displaystyle \L\,x\;=\;8\left(\sqrt{\frac{y}{7}}\right)^3\,-\,8\left(\sqrt{\frac{y}{7}}\right) \:=\:\frac{8y^{\frac{3}{2}}}{7\sqrt{7}}\,-\,\frac{8y^{\frac{1}{2}}}{\sqrt{7}} \:=\:\frac{8}{7\sqrt{7}}\left(y^{\frac{3}{2}}\,-\,7y^{\frac{1}{2}}\right)\)
Then: \(\displaystyle \L\:A\;=\;\2\,\times\,\frac{8}{7\sqrt{7}}\int^{\;\;\;56}_0\left(y^{\frac{3}{2}}\,-\,7y^{\frac{1}{2}}\right)\,dy\)
. . \(\displaystyle \L A \;=\;\left\, \frac{16}{7\sqrt{7}}\left(\frac{2}{5}y^{\frac{5}{2}} \,-\,\frac{14}{3}y^{\frac{3}{2}}\right)\,\right|^{56}_0 \;= \;\left\,\frac{16}{7\sqrt{7}}\cdot\frac{2y^{\frac{3}{2}}}{15}\left(3y\,-\,35\right)\,\right|^{56}_0 \;=\;\left\,\frac{32}{105\sqrt{7}}y^{\frac{3}{2}}(3y\,-\,35)\,\right|^{56}_0\)
Can you finish it now?
Hello, thebenji!
Eliminate the parameter (as Galactus did):
From [1], we have: \(\displaystyle \L\,t \:=\:\sqrt{\frac{y}{7}}\)
Substitute into [2]: \(\displaystyle \L\,x\;=\;8\left(\sqrt{\frac{y}{7}}\right)^3\,-\,8\left(\sqrt{\frac{y}{7}}\right) \:=\:\frac{8y^{\frac{3}{2}}}{7\sqrt{7}}\,-\,\frac{8y^{\frac{1}{2}}}{\sqrt{7}} \:=\:\frac{8}{7\sqrt{7}}\left(y^{\frac{3}{2}}\,-\,7y^{\frac{1}{2}}\right)\)
Then: \(\displaystyle \L\:A\;=\;\2\,\times\,\frac{8}{7\sqrt{7}}\int^{\;\;\;56}_0\left(y^{\frac{3}{2}}\,-\,7y^{\frac{1}{2}}\right)\,dy\)
. . \(\displaystyle \L A \;=\;\left\, \frac{16}{7\sqrt{7}}\left(\frac{2}{5}y^{\frac{5}{2}} \,-\,\frac{14}{3}y^{\frac{3}{2}}\right)\,\right|^{56}_0 \;= \;\left\,\frac{16}{7\sqrt{7}}\cdot\frac{2y^{\frac{3}{2}}}{15}\left(3y\,-\,35\right)\,\right|^{56}_0 \;=\;\left\,\frac{32}{105\sqrt{7}}y^{\frac{3}{2}}(3y\,-\,35)\,\right|^{56}_0\)
Can you finish it now?