#### To determine

**To find:**

The area of the given region.

#### Answer

43

#### Explanation

**1) Concept:**

The fundamental theorem of calculus part 2: If f is continuous on a,b, then ∫abfxdx=Fb-Fa, where F is any antiderivative of f , that is, the function F such that F'=f .

**2) Formula:**

∫fx+gxdx=∫fxdx+∫gxdx

∫xndx=xn+1n+1+C

∫kdx=kx+C

**3) Given:**

The equation of the curve is f(y)=x=2y-y2. We need to integrate form y=0 to y=2

**4) Calculation:**

To find the area of the region enclosed by the given curve:

From the given graph, the curve x=fy=2y-y2 bounded between y=0 and y=2 is continuous on 0,2 then, by using the fundamental theorem of calculus part 2, the area of region enclosed by the given curve is

∫022y-y2dy

To evaluate this definite integral, use the addition property of integral and separate the integral.

∫022y-y2dy=∫022ydy-∫02y2dy

Using the power rule of integration,

=2·y2202-y3302

By using the fundamental theorem of calculus,

=22-233-0

By simplifying,

∫022y-y2dy=43

**Conclusion:**

Therefore, the area of the given region is 43