Find the Area Between the Curves y=x , y=3 square root of x

Problem

y=x,y=3√(,x)

Solution

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

x=3√(,x)

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

x2=9*x

x2−9*x=0

x*(x−9)=0

x=0,x=9

3. Determine the upper curve by testing a value in the interval (0,9) For x=1 3√(,1)=3 and x=1 Since 3>1 the curve y=3√(,x) is the upper boundary.

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

A=(∫_0^9)((3√(,x)−x)*d(x))

5. Integrate the expression using the power rule.

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

2*x(3/2)−(x2)/2

6. Evaluate the definite integral at the boundaries 0 and 9

A=[2*(9)(3/2)−9/2]−[2*(0)(3/2)−0/2]

A=[2*(27)−81/2]−0

A=54−40.5

A=13.5

Final Answer

(∫_0^9)((3√(,x)−x)*d(x))=27/2

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