Find the Area Between the Curves y=x , y=2 square root of x

Problem

y=x,y=2√(,x)

Solution

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

x=2√(,x)

2. Solve for x by squaring both sides of the equation.

x2=4*x

x2−4*x=0

x*(x−4)=0

x=0,x=4

3. Determine the upper function on the interval [0,4] by testing a value like x=1 Since 2√(,1)=2 and 1 = 1,t*h*e*ƒ*u*n*c*t*i*o*n = 2\sqrt{x}$ is the upper curve.

4. Set up the integral for the area A using the formula A=(∫_a^b)((ƒ(x)−g(x))*d(x))

A=(∫_0^4)((2√(,x)−x)*d(x))

5. Integrate the expression using the power rule.

A=[2⋅(x(3/2))/(3/2)−(x2)/2]40

A=[4/3*x(3/2)−(x2)/2]40

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

A=(4/3*(4)(3/2)−4/2)−(4/3*(0)(3/2)−0/2)

A=(4/3⋅8−16/2)−0

A=32/3−8

A=32/3−24/3

A=8/3

Final Answer

(∫_0^4)((2√(,x)−x)*d(x))=8/3

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