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

Problem

y=x,y=√(4,x)

Solution

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

x=√(4,x)

x4=x

x4−x=0

x*(x3−1)=0

x=0,x=1

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

√(4,0.5)≈0.84

0.84>0.5

√(4,x)≥x

3. Set up the integral for the area by subtracting the lower function from the upper function over the interval [0,1]

A=(∫_0^1)((√(4,x)−x)*d(x))

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

(∫_^)(xn*d(x))=(x(n+1))/(n+1)

A=[4/5*x(5/4)−1/2*x2]10

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

A=(4/5*(1)(5/4)−1/2*(1)2)−(4/5*(0)(5/4)−1/2*(0)2)

A=4/5−1/2

6. Simplify the result by finding a common denominator.

A=8/10−5/10

A=3/10

Final Answer

(∫_0^1)((√(4,x)−x)*d(x))=3/10

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