(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the minimum value of the area of the region under the curve ## y=4x - x^3 ## from ##x=a## to ##x=a+1##, for all ##a>0##. This problem is from Stewart's Calculus

2. Relevant equations

Finding the area under the curve....

3. The attempt at a solution

I can set up the equation for the area as

$$A = \int_a^{a+1} (4x-x^3) \; dx $$ Solving this, we get,

$$ A = (a+1)^2 \left[ 2 - \frac{(a+1)^2}{4} \right ] $$ And I need to maximize this function. I can plug ##\alpha = (a+1)^2 ## and the area becomes ##A = \alpha (2 - \frac{\alpha}{4}) ##. But this looks like an inverted parabola and it would have a maxima and a minima of zero area. Does that make sense ?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Find the minimum value of area under y = 4x - x^3

Tags:

Have something to add?

**Physics Forums | Science Articles, Homework Help, Discussion**