Find the Area Between the Curves y=81x , y=x^5 , x=0 , x=3

Problem

(∫_0^3)(|81*x−x5|*d(x))

Solution

1. Identify the functions and the interval of integration. The curves are ƒ(x)=81*x and g(x)=x5 on the interval [0,3]

2. Determine which function is greater on the interval by finding intersection points. Set 81*x=x5

x5−81*x=0

x*(x4−81)=0

x*(x2−9)*(x2+9)=0

x*(x−3)*(x+3)*(x2+9)=0

The intersection points are x=0 x=3 and x=−3

3. Test a value in the interval (0,3) to see which curve is on top. For x=1

ƒ(1)=81*(1)=81

g(1)=(1)5=1

Since 81>1 ƒ(x)≥g(x) on the interval [0,3]

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

A=(∫_0^3)((81*x−x5)*d(x))

5. Integrate the expression using the power rule.

(∫_^)((81*x−x5)*d(x))=(81*x2)/2−(x6)/6

6. Evaluate the definite integral at the boundaries 0 and 3

A=[(81*(3)2)/2−((3)6)/6]−[(81*(0)2)/2−((0)6)/6]

A=(81⋅9)/2−729/6

A=729/2−243/2

A=486/2

A=243

Final Answer

(∫_0^3)((81*x−x5)*d(x))=243

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