MathDB

A5) Find the maximum value of

\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx

for

0\leq y\leq 1

. I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted

f(y)

, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule we have

\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}

Now what?

A5