Calculus Examples

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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 using the AC method.
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Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Write the factored form using these integers.
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to and solve for .
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Set equal to .
Add to both sides of the equation.
Set equal to and solve for .
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Set equal to .
Subtract from 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.
Simplify each term.
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Apply the distributive property.
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 .
Combine and .
Since is constant with respect to , move out of the integral.
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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Raise to the power of .
Combine and .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
Raise to the power of .
Combine and .
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 .
Subtract from .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
Raise to the power of .
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 .
Raise to the power of .
Move the negative in front of the fraction.
Multiply by .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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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.
Simplify the numerator.
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Multiply by .
Multiply by .
Subtract from .
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