To find the exact area of the region bounded by the curves $x = y^2 - 2y$ and $x + y^2 = 4y$, we need to first set up the equations and identify the points of intersection of the two curves.

Step 1: Set up the equations

The given equations are:

1. $x = y^2 - 2y$
2. $x + y^2 = 4y$

Let's first rearrange the second equation to express $x$:

$x = 4y - y^2$

Step 2: Find the points of intersection

To find the points where the curves intersect, we set the two expressions for $x$ equal to each other:

$y^2 - 2y = 4y - y^2$

Now, simplify this equation:

$y^2 - 2y = 4y - y^2$ $y^2 + y^2 - 2y - 4y = 0$ $2y^2 - 6y = 0$ $2y(y - 3) = 0$

So, the solutions are $y = 0$ and $y = 3$. These are the $y$-coordinates of the points of intersection.

Step 3: Set up the integral for the area

The area between two curves is given by the integral of the difference between the curves over the interval of intersection. The limits of integration are from $y = 0$ to $y = 3$.

The area is:

$A = \int_0^3 \left[ (4y - y^2) - (y^2 - 2y) \right] \, dy$

Simplify the integrand:

$A = \int_0^3 \left[ 4y - y^2 - y^2 + 2y \right] \, dy$ $A = \int_0^3 \left[ 6y - 2y^2 \right] \, dy$

Step 4: Compute the integral

Now, let's compute the integral:

$A = \int_0^3 6y \, dy - \int_0^3 2y^2 \, dy$

First, compute the integral of $6y$:

$\int 6y \, dy = 3y^2$

Now, compute the integral of $2y^2$:

$\int 2y^2 \, dy = \frac{2}{3}y^3$

Now, evaluate each of these integrals from 0 to 3:

[ A = \left[ 3y^2 \right]_0^3 - \left[ \frac{2}{3}y^3 \right]_0^3 ]

Evaluate:

$A = \left( 3(3^2) - 3(0^2) \right) - \left( \frac{2}{3}(3^3) - \frac{2}{3}(0^3) \right)$ $A = (3 \times 9) - \left( \frac{2}{3} \times 27 \right)$ $A = 27 - 18$ $A = 9$

Final Answer:

The exact area of the region bounded by the given curves is 9 square units.

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Would you like to go over any part of the solution in more detail or have any additional questions?

Here are some related questions you might find interesting:

1. How do you find the area between curves when one of them is horizontal?
2. What if the curves intersect at more than two points? How would that change the approach?
3. How do you determine if the curves are above or below each other in a given interval?
4. What is the general formula for finding the area between two curves?
5. How would you approach the area between curves in polar coordinates?

Tip: Always simplify your expressions and set the equations equal to each other first to find the points of intersection before setting up your integral. This step is key to getting the correct limits of integration!