integral and its largest value

nepenthe

hello..could you please help me to solve this problem?

Along what curve of the family y=x^n does the integral
int{(25xy-8y^2)dx} attain its largest value? and the boundaries for the integral is from (0,0) to (1,1)

thank you..

Galileo

Any thoughts on this problem? Maybe just evaluating the integral along the curve x^n?

HallsofIvy

\ doesn't mean a whole lot to me. Do you mean "where does x^n intersect the integral of 26xy- 8y^2 dx with the largest y value?

bomba923

Is this what you're looking for ??

[tex] \begin{gathered}
y = x^n \Rightarrow 25xy - 8y^2 = 25x^{n + 1} - 8x^{2n} \hfill \\
\frac{d}
{{dn}}\left[ {\int\limits_0^1 {\left( {25x^{n + 1} - 8x^{2n} } \right)dx} } \right] = \frac{{16}}{{\left( {2n + 1} \right)^2 }} - \frac{{25}}{{\left( {n + 2} \right)^2 }} = 0 \Rightarrow \frac{4}{{2n + 1}} = \frac{5}{{n + 2}} \Rightarrow n = \frac{1}
{2} \hfill \\
\therefore {\text{Curve is }}y = \sqrt x \hfill \\
\end{gathered} [/tex]

(If you allow 'n' to be a rational number, that is )

---?Though I'm not sure this is what you're looking for ?