$$\text{Evaluate by changing the order} \int^1_0\int^y_{4y}e^{x^{2}}dx\ dy.$$
I am unable to solve the following question. I have tried using the following approach. First I formed the equation of lines using the limits of the inner integral which gave equations of $2$ lines.
$$ y = x $$
$$ y = x/4 $$
and the limits of the outer integral gave me the total bounded region as below
From what I understood, I have to calculate the volume of function $e^{x^2}$ within the bounded area. Since, from the current order, calculating the integral is difficult, I tried to change the order of integral (as given in question) so this is what I did :
First I thought the current integral divides the current region into small $dA$ and first integrating by $dx$ means that we are taking a strip parallel to $X$-axis with length between $y = x$ and $y = x/4$ and then integrated it all the way above from 0 to 1.
Then I tried to change this order and thought of integrating first w.r.t. $dy$ as this will mean I will divide the region into strips parallel to $Y$-axis but the equation will be divided into two with the inner limits of first being $x/4$ to $x$ and second from $x/4$ to $1$. and the outer limit will change to 0 to 1 for first and 1 to 4 for second. I was solving the integral but then I encountered a problem.
I am unable to integrate the highlighted term any further. Please tell me where I went wrong.
Note: I forgot the $x$ in the first term, it will be $\frac{3xe^{x^2}}{4}$
$$ \Big[ y e^{x^2} \Big]_{y\,:=\,x/4}^{y\,:=\,x} = xe^{x^2} - \frac x4e^{x^2} = \frac3 4 xe^{x^2}. $$