Find the Area Between the Curves y=x^(5/4) , y=3x^(1/4)

Problem

y=x^(5/4),y=3*x^(1/4)

Solution

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

x^(5/4)=3*x^(1/4)

2. Solve for x by subtracting 3*x^(1/4) from both sides and factoring.

x^(5/4)−3*x^(1/4)=0

x^(1/4)*(x−3)=0

x=0,x=3

3. Determine the upper curve on the interval [0,3] by testing a point such as x=1 Since 3*(1)^(1/4)=3 and (1)^(5/4)=1 the curve y=3*x^(1/4) is the upper boundary.

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

A=(∫_0^3)((3*x^(1/4)−x^(5/4))*d(x))

5. Integrate each term using the power rule (∫_^)(x^n*d(x))=(x^(n+1))/(n+1)

A=[3⋅(x^(5/4))/(5/4)−(x^(9/4))/(9/4)]^3_0

A=[(12*x^(5/4))/5−(4*x^(9/4))/9]^3_0

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

A=((12*(3)^(5/4))/5−(4*(3)^(9/4))/9)−0

7. Simplify the expression by factoring out common terms. Note that 3^(9/4)=3^2⋅3^(1/4)=9⋅3^(1/4) and 3^(5/4)=3⋅3^(1/4)

A=(12*(3⋅3^(1/4)))/5−(4*(9⋅3^(1/4)))/9

A=(36⋅3^(1/4))/5−4⋅3^(1/4)

A=(36/5−20/5)⋅3^(1/4)

A=(16⋅3^(1/4))/5

Final Answer

(∫_0^3)((3*x^(1/4)−x^(5/4))*d(x))=(16⋅3^(1/4))/5

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