Find the total area of the region between the curve and the x-axis.
-$$y = x ^ { 2 } ( x - 2 ) ^ { 2 } ; 0 \leq x \leq 2$$
A) $\frac { 15 } { 17 }$
B) $\frac { 16 } { 15 }$
C) $\frac { 15 } { 16 }$
D) $\frac { 17 } { 15 }$

Question

Quizplus · Accepted Answer

To find the total area of the region between the curve $y = x^2(x - 2)^2$ and the x-axis from $x = 0$ to $x = 2$, we integrate the function with respect to $x$ over the given interval. The function is already non-negative in this interval, so we can directly integrate without worrying about absolute values.$$\text{Area} = \int_{0}^{2} x^2(x - 2)^2 dx$$Expanding the integrand gives:$$\int_{0}^{2} (x^4 - 4x^3 + 4x^2) dx$$Integrating term by term:$$= \left[ \frac{x^5}{5} - x^4 + \frac{4x^3}{3} \right]_0^2$$Evaluating the definite integral:$$= \left( \frac{32}{5} - 16 + \frac{32}{3} \right) - 0$$$$= \frac{32}{5} - 16 + \frac{32}{3}$$$$= \frac{96 + 160 - 240}{15}$$$$= \frac{16}{15}$$Therefore, the total area of the region between the curve and the x-axis is $\frac{16}{15}$.

Find the Total Area of the Region Between the Curve y=x2(x−2)2;0≤x≤2y = x ^ { 2 } ( x - 2 ) ^ { 2 } ; 0 \leq x \leq 2y=x2(x−2)2;0≤x≤2

Find the Total Area of the Region Between the Curve $y = x ^ { 2 } ( x - 2 ) ^ { 2 } ; 0 \leq x \leq 2$