In integration by substitution, we change the variable. For example: $$\int_{x^{2}=0}^{x^{2}=9} x^{2} d(x^{2})=\int_{x=0}^{x=3} x^{2} \dfrac{d(x^{2})}{dx}dx$$ Here since we have two different variables I find it impossible to graphically visualize how integration by substitution works.
Is there any other way to visualize this. Any help will be appreciated.
The plot below shows, in a Cartesian $(x, u, y)$-space, the change-of-variables identity $$ \int_{x = a}^{x = b} f(g(x)) g'(x)\, dx = \int_{u = g(a)}^{u = g(b)} f(u)\, du \tag{1} $$ for the region $R$ defined by $0 \leq y \leq f(u)$ for $g(a) \leq u \leq g(b)$ in the $(u, y)$-plane.
The integral on the left side of (1) represents the parametrization of $R$ by the region $R_{0}$ defined by $0 \leq y \leq f(x)$ for $a \leq x \leq b$ in the $(x, y)$-plane. To get the area of $R$ as an integral over $R_{0}$, send the point $(x, y)$ to $(u, y) = (g(x), y)$. As the shaded areas indicate, the vertical strip of $R_{0}$ at location $x$ is scaled in width (hence in area) by $g'(x)$.