find the total area of the shaded region.

Explanation:

Step 1: Determine the bounds of integration

The curves intersect at ( y = 0 ) and ( y = 1 ) (from point ((5,1))).

Step 2: Set up the integral for the area between curves

The right curve is ( x = 5y ), the left curve is ( x = 5y^2 ). Area formula:
$$ \int_{0}^{1} (5y - 5y^2) , dy $$

Step 3: Integrate term by term

Integrate ( 5y ) and ( 5y^2 ):
$$ \left[ \frac{5}{2}y^2 - \frac{5}{3}y^3 \right]_{0}^{1} $$

Step 4: Evaluate at bounds

Substitute ( y = 1 ) and ( y = 0 ):
$$ \left( \frac{5}{2} - \frac{5}{3} \right) - 0 = \frac{15}{6} - \frac{10}{6} = \frac{5}{6} $$

Answer:

(\frac{5}{6})