Question

Calculate the arithmetic average of the function on the D region and write its simplest form

y 4 - ezty fonksiyonunun D= {(x, y) € R²|0<I<1,0<y<1-x} bölgesi üzerindeki aritmetik ortalamasını hesaplayıp en sade halde yazınız.

Answer 1

We first find the area of D given by

$A= \int_{0}^{1}\int_{0}^{1-x}dydx \\ ~~~~~~~~=\int_{0}^{1}(1-x)dx\\ ~~~~~~~~=\left ( x-{x^2\over 2} \right )_{0}^{1}\\ ~~~~~~~~={1\over 2}$

Also,

Value of the function over the region is given by

$V= \int_{0}^{1}\int_{0}^{1-x}e^{{y\over y+x}} dy dx={1\over 2}(e-1)$

Therefore, the average value is

$\text{Average}={V\over A}= e-1$