Problem
der the Curve Find the exact area under the curve between the indicated values of $x$ $y=x^{2}(x-2)^{2}$, between $x=0$ and $x=2$ A. $\frac{16}{15}$ B. $\frac{15}{16}$ c. $\frac{17}{15}$ D. $\frac{15}{17}$
Solution
Step 1 ：We are given the function \(y=x^{2}(x-2)^{2}\) and we are asked to find the exact area under the curve between \(x=0\) and \(x=2\).
Step 2 ：This is a basic application of the Fundamental Theorem of Calculus, which states that the exact area under a curve \(y=f(x)\) from \(x=a\) to \(x=b\) is given by the definite integral \(\int_{a}^{b} f(x) dx\).
Step 3 ：Applying this theorem to our problem, we need to evaluate the definite integral \(\int_{0}^{2} x^{2}(x-2)^{2} dx\).
Step 4 ：Evaluating this integral, we find that the exact area under the curve is \(\frac{16}{15}\).
Step 5 ：Final Answer: The exact area under the curve \(y=x^{2}(x-2)^{2}\), between \(x=0\) and \(x=2\) is \(\boxed{\frac{16}{15}}\).
