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

Problem

y=x,y=√(6,x)

Solution

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

x=√(6,x)

x6=x

x6−x=0

x*(x5−1)=0

x=0,x=1

2. Determine the upper function on the interval [0,1] by testing a value such as x=0.5

√(6,0.5)≈0.89

0.89>0.5

√(6,x)≥x

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

A=(∫_0^1)(√(6,x)−x*d(x))

4. Apply the power rule for integration to each term.

A=[(x(7/6))/(7/6)−(x2)/2]10

A=[(6*x(7/6))/7−(x2)/2]10

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

A=((6*(1)(7/6))/7−((1)2)/2)−((6*(0)(7/6))/7−((0)2)/2)

A=6/7−1/2

6. Find a common denominator to subtract the fractions.

A=12/14−7/14

A=5/14

Final Answer

(∫_0^1)(√(6,x)−x*d(x))=5/14

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