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Calculus Examples
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Step 1
Step 1.1
Eliminate the equal sides of each equation and combine.
Step 1.2
Solve for .
Step 1.2.1
Subtract from both sides of the equation.
Step 1.2.2
Subtract from both sides of the equation.
Step 1.2.3
Subtract from .
Step 1.2.4
Factor the left side of the equation.
Step 1.2.4.1
Factor out of .
Step 1.2.4.1.1
Factor out of .
Step 1.2.4.1.2
Factor out of .
Step 1.2.4.1.3
Rewrite as .
Step 1.2.4.1.4
Factor out of .
Step 1.2.4.1.5
Factor out of .
Step 1.2.4.2
Factor.
Step 1.2.4.2.1
Factor using the AC method.
Step 1.2.4.2.1.1
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 .
Step 1.2.4.2.1.2
Write the factored form using these integers.
Step 1.2.4.2.2
Remove unnecessary parentheses.
Step 1.2.5
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.6
Set equal to and solve for .
Step 1.2.6.1
Set equal to .
Step 1.2.6.2
Add to both sides of the equation.
Step 1.2.7
Set equal to and solve for .
Step 1.2.7.1
Set equal to .
Step 1.2.7.2
Subtract from both sides of the equation.
Step 1.2.8
The final solution is all the values that make true.
Step 1.3
Evaluate when .
Step 1.3.1
Substitute for .
Step 1.3.2
Substitute for in and solve for .
Step 1.3.2.1
Remove parentheses.
Step 1.3.2.2
Remove parentheses.
Step 1.3.2.3
Add and .
Step 1.4
Evaluate when .
Step 1.4.1
Substitute for .
Step 1.4.2
Substitute for in and solve for .
Step 1.4.2.1
Remove parentheses.
Step 1.4.2.2
Remove parentheses.
Step 1.4.2.3
Add and .
Step 1.5
The solution to the system is the complete set of ordered pairs that are valid solutions.
Step 2
Reorder and .
Step 3
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 4
Step 4.1
Combine the integrals into a single integral.
Step 4.2
Simplify each term.
Step 4.2.1
Apply the distributive property.
Step 4.2.2
Multiply by .
Step 4.3
Subtract from .
Step 4.4
Split the single integral into multiple integrals.
Step 4.5
Since is constant with respect to , move out of the integral.
Step 4.6
By the Power Rule, the integral of with respect to is .
Step 4.7
Combine and .
Step 4.8
Since is constant with respect to , move out of the integral.
Step 4.9
By the Power Rule, the integral of with respect to is .
Step 4.10
Combine and .
Step 4.11
Apply the constant rule.
Step 4.12
Substitute and simplify.
Step 4.12.1
Evaluate at and at .
Step 4.12.2
Evaluate at and at .
Step 4.12.3
Evaluate at and at .
Step 4.12.4
Simplify.
Step 4.12.4.1
One to any power is one.
Step 4.12.4.2
Raise to the power of .
Step 4.12.4.3
Move the negative in front of the fraction.
Step 4.12.4.4
Multiply by .
Step 4.12.4.5
Multiply by .
Step 4.12.4.6
Combine the numerators over the common denominator.
Step 4.12.4.7
Add and .
Step 4.12.4.8
Cancel the common factor of and .
Step 4.12.4.8.1
Factor out of .
Step 4.12.4.8.2
Cancel the common factors.
Step 4.12.4.8.2.1
Factor out of .
Step 4.12.4.8.2.2
Cancel the common factor.
Step 4.12.4.8.2.3
Rewrite the expression.
Step 4.12.4.8.2.4
Divide by .
Step 4.12.4.9
Multiply by .
Step 4.12.4.10
One to any power is one.
Step 4.12.4.11
Raise to the power of .
Step 4.12.4.12
Cancel the common factor of and .
Step 4.12.4.12.1
Factor out of .
Step 4.12.4.12.2
Cancel the common factors.
Step 4.12.4.12.2.1
Factor out of .
Step 4.12.4.12.2.2
Cancel the common factor.
Step 4.12.4.12.2.3
Rewrite the expression.
Step 4.12.4.12.2.4
Divide by .
Step 4.12.4.13
Multiply by .
Step 4.12.4.14
To write as a fraction with a common denominator, multiply by .
Step 4.12.4.15
Combine and .
Step 4.12.4.16
Combine the numerators over the common denominator.
Step 4.12.4.17
Simplify the numerator.
Step 4.12.4.17.1
Multiply by .
Step 4.12.4.17.2
Subtract from .
Step 4.12.4.18
Move the negative in front of the fraction.
Step 4.12.4.19
Multiply by .
Step 4.12.4.20
Multiply by .
Step 4.12.4.21
To write as a fraction with a common denominator, multiply by .
Step 4.12.4.22
Combine and .
Step 4.12.4.23
Combine the numerators over the common denominator.
Step 4.12.4.24
Simplify the numerator.
Step 4.12.4.24.1
Multiply by .
Step 4.12.4.24.2
Add and .
Step 4.12.4.25
Move the negative in front of the fraction.
Step 4.12.4.26
Multiply by .
Step 4.12.4.27
Multiply by .
Step 4.12.4.28
Add and .
Step 4.12.4.29
To write as a fraction with a common denominator, multiply by .
Step 4.12.4.30
Combine and .
Step 4.12.4.31
Combine the numerators over the common denominator.
Step 4.12.4.32
Simplify the numerator.
Step 4.12.4.32.1
Multiply by .
Step 4.12.4.32.2
Add and .
Step 5