
Two students consider substitution geometrically.

Check out this dialogue between two calculus students (based on a true story):
Devyn
Riley! We should be able to figure some integrals geometrically using transformations of functions.
Riley
That sounds like a cool idea. Maybe, since we know the graph of $$ is a semicircle, we get an ellipse defined on $$ just by stretching the graph of $$ by a factor of $$ horizontally. The equation of this ellipse would be
Devyn
Exactly! So since we know that geometrically…
Riley
And we know that the area under $$ from $$ is twice the under $$
Devyn and Riley
We must have
Devyn and Riley
Jinx!
Devyn
It is kind of like we just stretched out our whole coordinate system, and that helped us solve an integral.
Riley
In this case, everything got stretched out by a constant factor of $$ in the horizontal direction. I wonder if we could ever say anything useful about cases where we stretch the $$-axis by a different amount at each point?
Devyn
Whao, that is a wild thought. That seems really hard. Since derivatives measure how much a function stretches a little piece of the domain, maybe the derivative will come into play here?
Riley
Say we know that Then, using this transformation idea, we can evaluate if $$ and $$. The value of the integral on this interval is $$.