where is the problem in following integration with substitution? A basic query

Question

I am interested in solving the following integration $$\int_0^t\int_0^{t}\cdots \int_0^{t}(x_1+x_2+\cdots x_M) dx_1 dx_2\cdots dx_M$$ Now if I use following substitution $$y_1=x_1+x_2\cdots x_M\\ y_2=x_2\\ \vdots \\y_M=x_M$$ then the Jacobian of this transformation will be $1$ and the limits of $y_1$ will go from $0\to Mt$, of $y_2$ from $0\to t$, and similarly of $y_M$ from $0\to t$. The integral after substitution will become $$\int_0^t\int_0^t \cdots \int_0^t \int_0^{Mt}y_1dy_1 dy_2 \cdots dy_M$$ $$\int_0^{Mt}y_1dy_1\int_0^{t}dy_2\cdots \int_0^{t}dy_M$$ $$\frac{(Mt)^2}{2}\times t^{M-1}$$ however this answer is different from what we get through simple integration without any substitution (the answer achieved through simple integration is $\frac{Mt^2}{2}\times t^{M-1}$). Is there something terribly wrong with what I am doing? I will be very thankful if somebody could help me in understanding the problem with this approach. Thanks in advance.

Answer 1

You have to work more on the domain. For example, if $t=3$ and $M=2$ the domain $0<x_1<3$ and $0<x_2<3$ transforms into the domain $D$ satisfying the inequalities $0<y_1-y_2<3$ and $0<y_2<3$ which looks like this:

If we want to use this, then $$ \iint_D y_1\,dy_1\,dy_2=\int_0^3\int_{y_2}^{y_2+3}y_1\,dy_1\,dy_2=\int_0^3\bigl[\frac{1}{2}(y_2+3)^2-y_2^2\bigr]\,dy_2=27. $$

I leave it to you to handle the general case.