Calculus Examples

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Solve by substitution to find the intersection between the curves.
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Eliminate the equal sides of each equation and combine.
Solve for .
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Subtract from both sides of the equation.
Factor the left side of the equation.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor.
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Factor by grouping.
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Reorder terms.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Factor out of .
Rewrite as plus
Apply the distributive property.
Multiply by .
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
Remove unnecessary parentheses.
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 .
Solve for .
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Subtract from both sides of the equation.
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Move the negative in front of the fraction.
Set equal to and solve for .
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Set equal to .
Add to both sides of the equation.
The final solution is all the values that make true.
Evaluate when .
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Substitute for .
Simplify .
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Multiply .
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Multiply by .
Combine and .
Move the negative in front of the fraction.
Evaluate when .
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Substitute for .
Multiply by .
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 .
Multiply by .
Split the single integral into multiple integrals.
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.
By the Power Rule, the integral of with respect to is .
Combine and .
Apply the constant rule.
Substitute and simplify.
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Evaluate at and at .
Evaluate at and at .
Evaluate at and at .
Simplify.
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One to any power is one.
Factor out of .
Apply the product rule to .
Raise to the power of .
Multiply by .
One to any power is one.
Factor out of .
Apply the product rule to .
Raise to the power of .
Move the negative in front of the fraction.
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Combine and .
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 .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Evaluate.
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Simplify the numerator.
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Apply the product rule to .
One to any power is one.
Raise to the power of .
Simplify the numerator.
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Apply the product rule to .
One to any power is one.
Raise to the power of .
Simplify the numerator.
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Combine the numerators over the common denominator.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Subtract from .
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply by .
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.
Combine the numerators over the common denominator.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Add and .
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Multiply by .
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.
Move the negative in front of the fraction.
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 .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Multiply the numerator by the reciprocal of the denominator.
Multiply .
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Multiply and .
Multiply by .
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