Calculus Examples

Find the Area Between the Curves y=x , y=x^2
,
Solve by substitution to find the intersection between the curves.
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Substitute for into then solve for .
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Replace with in the equation.
Solve the equation for .
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Subtract from both sides of the equation.
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to .
Set the next factor equal to and solve.
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Set the next factor equal to .
Add to both sides of the equation.
The final solution is all the values that make true.
Substitute for into then solve for .
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Replace with in the equation.
Remove parentheses.
Substitute for into then solve for .
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Replace with in the equation.
Remove parentheses.
The solution to the system is the complete set of ordered pairs that are valid solutions.
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.
Integrate to find the area between and .
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Combine the integrals into a single integral.
Multiply by .
Split the single integral into multiple integrals.
By the Power Rule, the integral of with respect to is .
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
Simplify the answer.
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Combine and .
Substitute and simplify.
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Evaluate at and at .
Evaluate at and at .
Simplify.
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One to any power is one.
Multiply by .
Raising to any positive power yields .
Multiply by .
Multiply by .
Add and .
One to any power is one.
Raising to any positive power yields .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Add and .
To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Subtract from .
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