Brain Teaser - Area enclosed by a parabola and a line

[Gerson W. Barbosa]

(09-15-2015 02:44 PM)CR Haeger Wrote:  **or an old student looking at this again after a few decades of rust.

In this case Calculus Refresher, by A. A. Klaf might be handy. I bought this book last year in a local bookstore. But don't do like me, who kept it the shelf until today.
The equation of the normal line shouldn't be difficult to derive, but it's ready for use on page 70:

\[y-y_{1}=-\frac{1}{\frac{dy}{dx}}(x-x_{1})\]

For the case y = x^4, we have

\(y-y_{1}=-\frac{1}{4x_{1}^{3}}(x-x_{1})\)
\((x_{1},y_{1})=(u,u^{4})\)
\(y-u^{4}=-\frac{1}{4u^{3}}(x-u)\)
\(y=u^{4}-\frac{x-u}{4u^{3}}\)
\(y=\frac{4u^{7}+u-x}{4u^{3}}\)

Thanks for posting this interesting problem!

Gerson.


Edited to fix a typo and to add a missing subscript (Thanks, Thomas!).