Find the total area of the shaded region bounded by the following curves.
\[
\begin{array}{l}
x=36 y^{2}-12 y^{3} \\
x=6 y^{2}-18 y
\end{array}
\]
Answer
Final Answer: The total area of the shaded region bounded by the curves \(x=36 y^{2}-12 y^{3}\) and \(x=6 y^{2}-18 y\) is \(\boxed{\frac{13}{16}}\).
Step 1 ：We are given two curves, \(x=36 y^{2}-12 y^{3}\) and \(x=6 y^{2}-18 y\).
Step 2 ：The area between two curves is given by the integral of the absolute difference between the two functions.
Step 3 ：We first need to find the points of intersection of the two curves to determine the limits of integration.
Step 4 ：The points of intersection are \(-\frac{1}{2}\), 0, and 3.
Step 5 ：By integrating the absolute difference between the two functions over these limits, we find that the total area of the shaded region is \(\frac{13}{16}\).
Step 6 ：Final Answer: The total area of the shaded region bounded by the curves \(x=36 y^{2}-12 y^{3}\) and \(x=6 y^{2}-18 y\) is \(\boxed{\frac{13}{16}}\).
