Calculus Examples

Find the Area Between the Curves y=x , y=x^3 , x=0 , x=1
, , ,
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
Raise to the power of .
Step 1.2.2.1.2
Factor out of .
Step 1.2.2.1.3
Factor out of .
Step 1.2.2.1.4
Factor out of .
Step 1.2.2.2
Rewrite as .
Step 1.2.2.3
Factor.
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Step 1.2.2.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.2.2.3.2
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
Subtract from both sides of the equation.
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
Solve for .
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Step 1.2.6.2.1
Subtract from both sides of the equation.
Step 1.2.6.2.2
Divide each term in by and simplify.
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Step 1.2.6.2.2.1
Divide each term in by .
Step 1.2.6.2.2.2
Simplify the left side.
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Step 1.2.6.2.2.2.1
Dividing two negative values results in a positive value.
Step 1.2.6.2.2.2.2
Divide by .
Step 1.2.6.2.2.3
Simplify the right side.
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Step 1.2.6.2.2.3.1
Divide by .
Step 1.2.7
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
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
Raise to the power of .
Step 1.5
Evaluate when .
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Step 1.5.1
Substitute for .
Step 1.5.2
Substitute for in and solve for .
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Step 1.5.2.1
Remove parentheses.
Step 1.5.2.2
One to any power is one.
Step 1.6
The solution to the system is the complete set of ordered pairs that are valid 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
By the Power Rule, the integral of with respect to is .
Step 3.5
Since is constant with respect to , move out of the integral.
Step 3.6
By the Power Rule, the integral of with respect to is .
Step 3.7
Simplify the answer.
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Step 3.7.1
Combine and .
Step 3.7.2
Substitute and simplify.
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Step 3.7.2.1
Evaluate at and at .
Step 3.7.2.2
Evaluate at and at .
Step 3.7.2.3
Simplify.
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Step 3.7.2.3.1
One to any power is one.
Step 3.7.2.3.2
Multiply by .
Step 3.7.2.3.3
Raising to any positive power yields .
Step 3.7.2.3.4
Multiply by .
Step 3.7.2.3.5
Multiply by .
Step 3.7.2.3.6
Add and .
Step 3.7.2.3.7
One to any power is one.
Step 3.7.2.3.8
Raising to any positive power yields .
Step 3.7.2.3.9
Cancel the common factor of and .
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Step 3.7.2.3.9.1
Factor out of .
Step 3.7.2.3.9.2
Cancel the common factors.
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Step 3.7.2.3.9.2.1
Factor out of .
Step 3.7.2.3.9.2.2
Cancel the common factor.
Step 3.7.2.3.9.2.3
Rewrite the expression.
Step 3.7.2.3.9.2.4
Divide by .
Step 3.7.2.3.10
Multiply by .
Step 3.7.2.3.11
Add and .
Step 3.7.2.3.12
To write as a fraction with a common denominator, multiply by .
Step 3.7.2.3.13
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 3.7.2.3.13.1
Multiply by .
Step 3.7.2.3.13.2
Multiply by .
Step 3.7.2.3.14
Combine the numerators over the common denominator.
Step 3.7.2.3.15
Subtract from .
Step 4