   Chapter 4.3, Problem 42E

Chapter
Section
Textbook Problem

# Sketch the region enclosed by the given curves and calculate its area. y = 2 x − x 2 ,     y = 0

To determine

To sketch:

The region enclosed by given curves and to calculate its area.

Explanation

1) Concept:

i) Fundamental theorem of Calculus, Part 2

If f is continuous on a,b, then

abfxdx=Fb-F(a)

where F is any antiderivative of f, that is, a function F such that F'=f

ii) Power rule for antiderivative:

ddxxn+1n+1=xn

2) Given:

y=2x-x2, y=0

3) Calculation:

The curves  y=2x-x2, y=0 are shown below

The region enclosed by the curve y=2x-x2 , y=0 is shaded in black

To find the  area of the region enclosed by the given curves

From above graph, the curve y=fx=2x-x2 bounded between x=0 and x=2, is continuous on 0,2, then

By using concept i) (Fundamental theorem of Calculus, Part 2),

02(2x-x2) dx=F2-F0(1)

Where F is antiderivative of f, that is, a function F such that F'(x)=f(x) means

ddxFx

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