Find the total area of the region between the curve and the x-axis.
-$$y = 2 x - x ^ { 2 } ; 0 \leq x \leq 2$$
A) $\frac { 4 } { 3 }$
B) $\frac { 7 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 5 } { 3 }$

Question

Quizplus · Accepted Answer

The area between the curve and the x-axis is given by the integral of the absolute value of the function over the given interval. In this case, the function is $y = 2 x - x ^ { 2 } $, and the interval is $0 \leq x \leq 2$. So, the area is$$\int_{0}^{2} |2 x - x ^ { 2 } | dx = \int_{0}^{1} (2 x - x ^ { 2 }) dx + \int_{1}^{2} (x ^ { 2 } - 2 x) dx = \frac { 4 } { 3 }$$

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

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