<s>[INST] Find $n$ if the area between the curve of $y=x^n$ and the $y$ axis is $3$ times the area between the curve and the $x$-axis 
For this question i tried to find the area for red and blue sections, and equate them by red= 3 blue. 
However, it didnt work out and I got $b-a = 3(b^n - a^n)$ for the outcome. 
In the book, it says we dont have to deal with the red area....
Thank you.
 [/INST] The blue area is $$\int_a^b x^n dx = \frac{b^{n+1}-a^{n+1}}{n+1}.$$
The red (orange?) area is the area of the big rectangle with sides, $b$, $b^n$, minus the small (white) rectangle with sides $a$, $a^n$, minus the above integral. That is $$b^{n+1}-a^{n+1}-\frac{b^{n+1}-a^{n+1}}{n+1}.$$ Hence, we need $$ b^{n+1}-a^{n+1}-\frac{b^{n+1}-a^{n+1}}{n+1} = 3 \cdot \frac{b^{n+1}-a^{n+1}}{n+1}. $$ Simplifying a bit gives $$ (n+1)(b^{n+1} - a^{n+1}) = 4 (b^{n+1} - a^{n+1}). $$ So we have $$(n - 3)(b^{n+1}-a^{n+1})=0.$$ Since $a$ and $b$ are arbitrary, we require $n=3$.
</s>