# Integral and its largest value

1. Dec 3, 2005

### nepenthe

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..

2. Dec 3, 2005

### Galileo

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

3. Dec 3, 2005

### HallsofIvy

Staff Emeritus
\ 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?

4. Dec 3, 2005

### bomba923

Is this what you're looking for ??

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

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

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

Last edited: Dec 3, 2005