Calculus Examples

Find the Area Between the Curves y=81x , y=x^5 , x=0 , x=3
, , ,
Step 1
Solve by substitution to find the intersection between the curves.
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Step 1.1
Eliminate the equal sides of each equation and combine.
Step 1.2
Solve for .
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Step 1.2.1
Subtract from both sides of the equation.
Step 1.2.2
Factor the left side of the equation.
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Step 1.2.2.1
Factor out of .
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Step 1.2.2.1.1
Factor out of .
Step 1.2.2.1.2
Factor out of .
Step 1.2.2.1.3
Factor out of .
Step 1.2.2.2
Rewrite as .
Step 1.2.2.3
Rewrite as .
Step 1.2.2.4
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.2.2.5
Factor.
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Step 1.2.2.5.1
Simplify.
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Step 1.2.2.5.1.1
Rewrite as .
Step 1.2.2.5.1.2
Factor.
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Step 1.2.2.5.1.2.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.2.2.5.1.2.2
Remove unnecessary parentheses.
Step 1.2.2.5.2
Remove unnecessary parentheses.
Step 1.2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4
Set equal to .
Step 1.2.5
Set equal to and solve for .
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Step 1.2.5.1
Set equal to .
Step 1.2.5.2
Solve for .
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Step 1.2.5.2.1
Subtract from both sides of the equation.
Step 1.2.5.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.5.2.3
Simplify .
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Step 1.2.5.2.3.1
Rewrite as .
Step 1.2.5.2.3.2
Rewrite as .
Step 1.2.5.2.3.3
Rewrite as .
Step 1.2.5.2.3.4
Rewrite as .
Step 1.2.5.2.3.5
Pull terms out from under the radical, assuming positive real numbers.
Step 1.2.5.2.3.6
Move to the left of .
Step 1.2.5.2.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.5.2.4.1
First, use the positive value of the to find the first solution.
Step 1.2.5.2.4.2
Next, use the negative value of the to find the second solution.
Step 1.2.5.2.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.2.6
Set equal to and solve for .
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Step 1.2.6.1
Set equal to .
Step 1.2.6.2
Subtract from both sides of the equation.
Step 1.2.7
Set equal to and solve for .
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Step 1.2.7.1
Set equal to .
Step 1.2.7.2
Solve for .
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Step 1.2.7.2.1
Subtract from both sides of the equation.
Step 1.2.7.2.2
Divide each term in by and simplify.
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Step 1.2.7.2.2.1
Divide each term in by .
Step 1.2.7.2.2.2
Simplify the left side.
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Step 1.2.7.2.2.2.1
Dividing two negative values results in a positive value.
Step 1.2.7.2.2.2.2
Divide by .
Step 1.2.7.2.2.3
Simplify the right side.
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Step 1.2.7.2.2.3.1
Divide by .
Step 1.2.8
The final solution is all the values that make true.
Step 1.3
Evaluate when .
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Step 1.3.1
Substitute for .
Step 1.3.2
Substitute for in and solve for .
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Step 1.3.2.1
Remove parentheses.
Step 1.3.2.2
Raising to any positive power yields .
Step 1.4
Evaluate when .
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Step 1.4.1
Substitute for .
Step 1.4.2
Simplify .
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Step 1.4.2.1
Apply the product rule to .
Step 1.4.2.2
Raise to the power of .
Step 1.4.2.3
Factor out .
Step 1.4.2.4
Rewrite as .
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Step 1.4.2.4.1
Rewrite as .
Step 1.4.2.4.2
Rewrite as .
Step 1.4.2.4.3
Raise to the power of .
Step 1.4.2.5
Multiply by .
Step 1.5
Evaluate when .
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Step 1.5.1
Substitute for .
Step 1.5.2
Simplify .
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Step 1.5.2.1
Apply the product rule to .
Step 1.5.2.2
Raise to the power of .
Step 1.5.2.3
Factor out .
Step 1.5.2.4
Rewrite as .
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Step 1.5.2.4.1
Rewrite as .
Step 1.5.2.4.2
Rewrite as .
Step 1.5.2.4.3
Raise to the power of .
Step 1.5.2.5
Multiply by .
Step 1.6
Evaluate when .
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Step 1.6.1
Substitute for .
Step 1.6.2
Raise to the power of .
Step 1.7
Evaluate when .
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Step 1.7.1
Substitute for .
Step 1.7.2
Substitute for in and solve for .
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Step 1.7.2.1
Remove parentheses.
Step 1.7.2.2
Raise to the power of .
Step 1.8
List all of the solutions.
Step 2
The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.
Step 3
Integrate to find the area between and .
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Step 3.1
Combine the integrals into a single integral.
Step 3.2
Multiply by .
Step 3.3
Split the single integral into multiple integrals.
Step 3.4
Since is constant with respect to , move out of the integral.
Step 3.5
By the Power Rule, the integral of with respect to is .
Step 3.6
Combine and .
Step 3.7
Since is constant with respect to , move out of the integral.
Step 3.8
By the Power Rule, the integral of with respect to is .
Step 3.9
Simplify the answer.
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Step 3.9.1
Combine and .
Step 3.9.2
Substitute and simplify.
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Step 3.9.2.1
Evaluate at and at .
Step 3.9.2.2
Evaluate at and at .
Step 3.9.2.3
Simplify.
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Step 3.9.2.3.1
Raise to the power of .
Step 3.9.2.3.2
Raising to any positive power yields .
Step 3.9.2.3.3
Cancel the common factor of and .
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Step 3.9.2.3.3.1
Factor out of .
Step 3.9.2.3.3.2
Cancel the common factors.
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Step 3.9.2.3.3.2.1
Factor out of .
Step 3.9.2.3.3.2.2
Cancel the common factor.
Step 3.9.2.3.3.2.3
Rewrite the expression.
Step 3.9.2.3.3.2.4
Divide by .
Step 3.9.2.3.4
Multiply by .
Step 3.9.2.3.5
Add and .
Step 3.9.2.3.6
Combine and .
Step 3.9.2.3.7
Multiply by .
Step 3.9.2.3.8
Raise to the power of .
Step 3.9.2.3.9
Cancel the common factor of and .
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Step 3.9.2.3.9.1
Factor out of .
Step 3.9.2.3.9.2
Cancel the common factors.
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Step 3.9.2.3.9.2.1
Factor out of .
Step 3.9.2.3.9.2.2
Cancel the common factor.
Step 3.9.2.3.9.2.3
Rewrite the expression.
Step 3.9.2.3.10
Raising to any positive power yields .
Step 3.9.2.3.11
Cancel the common factor of and .
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Step 3.9.2.3.11.1
Factor out of .
Step 3.9.2.3.11.2
Cancel the common factors.
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Step 3.9.2.3.11.2.1
Factor out of .
Step 3.9.2.3.11.2.2
Cancel the common factor.
Step 3.9.2.3.11.2.3
Rewrite the expression.
Step 3.9.2.3.11.2.4
Divide by .
Step 3.9.2.3.12
Multiply by .
Step 3.9.2.3.13
Add and .
Step 3.9.2.3.14
Combine the numerators over the common denominator.
Step 3.9.2.3.15
Subtract from .
Step 3.9.2.3.16
Cancel the common factor of and .
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Step 3.9.2.3.16.1
Factor out of .
Step 3.9.2.3.16.2
Cancel the common factors.
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Step 3.9.2.3.16.2.1
Factor out of .
Step 3.9.2.3.16.2.2
Cancel the common factor.
Step 3.9.2.3.16.2.3
Rewrite the expression.
Step 3.9.2.3.16.2.4
Divide by .
Step 4