Calculus Examples

Find the Area Between the Curves y=x , y = fourth root of x
,
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
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 1.2.2
To remove the radical on the left side of the equation, raise both sides of the equation to the power of .
Step 1.2.3
Simplify each side of the equation.
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Step 1.2.3.1
Use to rewrite as .
Step 1.2.3.2
Simplify the left side.
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Step 1.2.3.2.1
Simplify .
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Step 1.2.3.2.1.1
Multiply the exponents in .
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Step 1.2.3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 1.2.3.2.1.1.2
Cancel the common factor of .
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Step 1.2.3.2.1.1.2.1
Cancel the common factor.
Step 1.2.3.2.1.1.2.2
Rewrite the expression.
Step 1.2.3.2.1.2
Simplify.
Step 1.2.4
Solve for .
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Step 1.2.4.1
Subtract from both sides of the equation.
Step 1.2.4.2
Factor the left side of the equation.
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Step 1.2.4.2.1
Factor out of .
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Step 1.2.4.2.1.1
Raise to the power of .
Step 1.2.4.2.1.2
Factor out of .
Step 1.2.4.2.1.3
Factor out of .
Step 1.2.4.2.1.4
Factor out of .
Step 1.2.4.2.2
Rewrite as .
Step 1.2.4.2.3
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 1.2.4.2.4
Factor.
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Step 1.2.4.2.4.1
Simplify.
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Step 1.2.4.2.4.1.1
One to any power is one.
Step 1.2.4.2.4.1.2
Multiply by .
Step 1.2.4.2.4.2
Remove unnecessary parentheses.
Step 1.2.4.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4.4
Set equal to .
Step 1.2.4.5
Set equal to and solve for .
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Step 1.2.4.5.1
Set equal to .
Step 1.2.4.5.2
Solve for .
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Step 1.2.4.5.2.1
Subtract from both sides of the equation.
Step 1.2.4.5.2.2
Divide each term in by and simplify.
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Step 1.2.4.5.2.2.1
Divide each term in by .
Step 1.2.4.5.2.2.2
Simplify the left side.
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Step 1.2.4.5.2.2.2.1
Dividing two negative values results in a positive value.
Step 1.2.4.5.2.2.2.2
Divide by .
Step 1.2.4.5.2.2.3
Simplify the right side.
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Step 1.2.4.5.2.2.3.1
Divide by .
Step 1.2.4.6
Set equal to and solve for .
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Step 1.2.4.6.1
Set equal to .
Step 1.2.4.6.2
Solve for .
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Step 1.2.4.6.2.1
Use the quadratic formula to find the solutions.
Step 1.2.4.6.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 1.2.4.6.2.3
Simplify.
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Step 1.2.4.6.2.3.1
Simplify the numerator.
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Step 1.2.4.6.2.3.1.1
One to any power is one.
Step 1.2.4.6.2.3.1.2
Multiply .
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Step 1.2.4.6.2.3.1.2.1
Multiply by .
Step 1.2.4.6.2.3.1.2.2
Multiply by .
Step 1.2.4.6.2.3.1.3
Subtract from .
Step 1.2.4.6.2.3.1.4
Rewrite as .
Step 1.2.4.6.2.3.1.5
Rewrite as .
Step 1.2.4.6.2.3.1.6
Rewrite as .
Step 1.2.4.6.2.3.2
Multiply by .
Step 1.2.4.6.2.4
Simplify the expression to solve for the portion of the .
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Step 1.2.4.6.2.4.1
Simplify the numerator.
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Step 1.2.4.6.2.4.1.1
One to any power is one.
Step 1.2.4.6.2.4.1.2
Multiply .
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Step 1.2.4.6.2.4.1.2.1
Multiply by .
Step 1.2.4.6.2.4.1.2.2
Multiply by .
Step 1.2.4.6.2.4.1.3
Subtract from .
Step 1.2.4.6.2.4.1.4
Rewrite as .
Step 1.2.4.6.2.4.1.5
Rewrite as .
Step 1.2.4.6.2.4.1.6
Rewrite as .
Step 1.2.4.6.2.4.2
Multiply by .
Step 1.2.4.6.2.4.3
Change the to .
Step 1.2.4.6.2.4.4
Rewrite as .
Step 1.2.4.6.2.4.5
Factor out of .
Step 1.2.4.6.2.4.6
Factor out of .
Step 1.2.4.6.2.4.7
Move the negative in front of the fraction.
Step 1.2.4.6.2.5
Simplify the expression to solve for the portion of the .
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Step 1.2.4.6.2.5.1
Simplify the numerator.
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Step 1.2.4.6.2.5.1.1
One to any power is one.
Step 1.2.4.6.2.5.1.2
Multiply .
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Step 1.2.4.6.2.5.1.2.1
Multiply by .
Step 1.2.4.6.2.5.1.2.2
Multiply by .
Step 1.2.4.6.2.5.1.3
Subtract from .
Step 1.2.4.6.2.5.1.4
Rewrite as .
Step 1.2.4.6.2.5.1.5
Rewrite as .
Step 1.2.4.6.2.5.1.6
Rewrite as .
Step 1.2.4.6.2.5.2
Multiply by .
Step 1.2.4.6.2.5.3
Change the to .
Step 1.2.4.6.2.5.4
Rewrite as .
Step 1.2.4.6.2.5.5
Factor out of .
Step 1.2.4.6.2.5.6
Factor out of .
Step 1.2.4.6.2.5.7
Move the negative in front of the fraction.
Step 1.2.4.6.2.6
The final answer is the combination of both solutions.
Step 1.2.4.7
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
Simplify .
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Step 1.3.2.1
Remove parentheses.
Step 1.3.2.2
Rewrite as .
Step 1.3.2.3
Pull terms out from under the radical, assuming positive real numbers.
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
Remove parentheses.
Step 1.4.2.2
Any root of is .
Step 1.5
Evaluate when .
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Step 1.5.1
Substitute for .
Step 1.5.2
Remove parentheses.
Step 1.6
Evaluate when .
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Step 1.6.1
Substitute for .
Step 1.6.2
Remove parentheses.
Step 1.7
List all of the solutions.
Step 2
The area between the given curves is unbounded.
Unbounded area
Step 3