MediumChange of variables

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §14.2 Change to Better Coordinates, #25 · Multivariable Calculus · Change of variables

If … lies between … and … under the graph of …

If $R$ lies between $x = 0$ and $x = 1$ under the graph of $y = f (x) > 0$ , then $x = u$ , $y = v f (u)$ takes $R$ to the unit square $S$ . Locate the corners of $R$ and the point corresponding to $u = \frac{1}{2}$ , $v = 1$ . Compute $J$ to prove what we know:
$area of R = \int_{0}^{1} f (x) d x = \int_{0}^{1} \int_{0}^{1} J d u d v .$

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §14.2 Change to Better Coordinates, #25

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