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 (∫_^)(xn*d(x))=(x(n+1))/(n+1)

A=[3⋅(x(5/4))/(5/4)−(x(9/4))/(9/4)]30

A=[(12*x(5/4))/5−(4*x(9/4))/9]30

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⋅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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