How would you integrate something like
sqrt(9-y)*y dy from 0 to 9
The problem is that "9- y" inside the square root, right? So try u= 9- y. Then du= -dy. and, of course, y= 9- u. When y= 0, u= 9 and when y= 9, u= 0.
So the $\displaystyle \sqrt{9- y}y dy= \sqrt{u}(9- u)(-du)= -(9u^{1/2}- u^{3/2))du$ and the integral will be from 9 to 0. Swapping the limits of integration gets rid of that negative.
Hello, CoffeeBird!
$\displaystyle \int^9_0y\sqrt{9-y}\:dy$
If the expression under the radical is linear,
. . we can let $\displaystyle u$ equal the entire radical.
Let $\displaystyle u \:=\:\sqrt{9-y} \quad\Rightarrow\quad u^2 \:=\:9-y \quad\Rightarrow\quad y\:=\:9-u^2 \quad\Rightarrow\quad du \:=\:-2u\,du$
Substitute: .$\displaystyle \int (9-u^2)\cdot u \cdot (-2u\,du) \;=\; -2\int(9u^2 - u^4)\,du$
. . . . . . . . $\displaystyle =\;-2\left(3u^3 - \tfrac{1}{5}u^5\right) + C \;=\;-\tfrac{2}{5}u^3\left(15 - u^2\right) + C$
Back-substitute: .$\displaystyle -\tfrac{2}{5}(9-y)^{\frac{3}{2}}(15 - [9-y]) + C \;=\;-\tfrac{2}{5}(9-y)^{\frac{3}{2}}(6 + y) + C$
Evaluate: .$\displaystyle -\tfrac{2}{5}(9-y)^{\frac{3}{2}}(6+y)\,\bigg]^9_0 \;=\;-\tfrac{2}{5}(0)^{\frac{3}{2}}(15) \:+\:\tfrac{2}{5}(9)^{\frac{3}{2}}(6) $
. . . . . . . . $\displaystyle =\;0 + \tfrac{2}{5}(27)(6) \;=\;\frac{324}{5}$