Find the Area Between the Curves y=x , y = square root of x

Problem

y=x,y=√(,x)

Solution

1. Find the intersection points by setting the two equations equal to each other to determine the limits of integration.

x=√(,x)

x2=x

x2−x=0

x*(x−1)=0

x=0,x=1

2. Determine the upper curve on the interval [0,1] by testing a value such as x=0.25

√(,0.25)=0.5

0.5>0.25

√(,x)≥x

3. Set up the definite integral for the area A by subtracting the lower function from the upper function over the interval.

A=(∫_0^1)(√(,x)−x*d(x))

4. Find the antiderivative of the integrand using the power rule.

(∫_^)(x(1/2)−x*d(x))=(2*x(3/2))/3−(x2)/2

5. Evaluate the definite integral at the upper and lower limits.

A=((2*(1)(3/2))/3−((1)2)/2)−((2*(0)(3/2))/3−((0)2)/2)

A=2/3−1/2

A=4/6−3/6

A=1/6

Final Answer

(∫_0^1)(√(,x)−x*d(x))=1/6

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