Area between curves

• Jan 22nd 2009, 09:30 AM
Dana_Scully
Area between curves
Is there something I'm doing wrong in this problem? I can't figure it out, which probably means it's blindingly obvious.

Find the area of the region enclosed by $\displaystyle y=\sqrt{x}, y=\frac{1}{2}x,$ and $\displaystyle x=16$.

$\displaystyle \sqrt{x}=\frac{1}{2}x$ at $\displaystyle x=(0, 4)$

So $\displaystyle \displaystyle\int^{16}_4 \frac{1}{2}x - x^{\frac{1}{2}} dx$

$\displaystyle \left(\frac{x^2}{4} - \frac{2\sqrt{x^3}}{3}\right) \Bigg|_{4}^{16}$

$\displaystyle \left(64 - \frac{128}{3}\right) - \left(4 - \frac{16}{3}\right)$

Which equals $\displaystyle \frac{68}{3}$
• Jan 22nd 2009, 09:39 AM
ThePerfectHacker
Quote:

Originally Posted by Dana_Scully
Is there something I'm doing wrong in this problem? I can't figure it out, which probably means it's blindingly obvious.

Find the area of the region enclosed by $\displaystyle y=\sqrt{x}, y=\frac{1}{2}x,$ and $\displaystyle x=16$.

$\displaystyle \sqrt{x}=\frac{1}{2}x$ at $\displaystyle x=(0, 4)$

So $\displaystyle \displaystyle\int^{16}_4 \frac{1}{2}x - x^{\frac{1}{2}} dx$

$\displaystyle \left(\frac{x^2}{4} - \frac{2\sqrt{x^3}}{3}\right) \Bigg|_{4}^{16}$

$\displaystyle \left(64 - \frac{128}{3}\right) - \left(4 - \frac{16}{3}\right)$

Which equals $\displaystyle \frac{68}{3}$

Look at the picutre.
• Jan 22nd 2009, 09:54 AM
Dana_Scully
Thank you! I guess I thought that it meant only the part directly bounded by x=16...I'm new to this, so thanks.