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Calculus Examples
, ,
Step 1
Step 1.1
Eliminate the equal sides of each equation and combine.
Step 1.2
Solve for .
Step 1.2.1
Find the LCD of the terms in the equation.
Step 1.2.1.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 1.2.1.2
The LCM of one and any expression is the expression.
Step 1.2.2
Multiply each term in by to eliminate the fractions.
Step 1.2.2.1
Multiply each term in by .
Step 1.2.2.2
Simplify the left side.
Step 1.2.2.2.1
Cancel the common factor of .
Step 1.2.2.2.1.1
Cancel the common factor.
Step 1.2.2.2.1.2
Rewrite the expression.
Step 1.2.2.3
Simplify the right side.
Step 1.2.2.3.1
Multiply by by adding the exponents.
Step 1.2.2.3.1.1
Move .
Step 1.2.2.3.1.2
Multiply by .
Step 1.2.3
Solve the equation.
Step 1.2.3.1
Rewrite the equation as .
Step 1.2.3.2
Divide each term in by and simplify.
Step 1.2.3.2.1
Divide each term in by .
Step 1.2.3.2.2
Simplify the left side.
Step 1.2.3.2.2.1
Cancel the common factor of .
Step 1.2.3.2.2.1.1
Cancel the common factor.
Step 1.2.3.2.2.1.2
Divide by .
Step 1.2.3.2.3
Simplify the right side.
Step 1.2.3.2.3.1
Cancel the common factor of and .
Step 1.2.3.2.3.1.1
Factor out of .
Step 1.2.3.2.3.1.2
Cancel the common factors.
Step 1.2.3.2.3.1.2.1
Factor out of .
Step 1.2.3.2.3.1.2.2
Cancel the common factor.
Step 1.2.3.2.3.1.2.3
Rewrite the expression.
Step 1.2.3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.3.4
Simplify .
Step 1.2.3.4.1
Rewrite as .
Step 1.2.3.4.2
Any root of is .
Step 1.2.3.4.3
Simplify the denominator.
Step 1.2.3.4.3.1
Rewrite as .
Step 1.2.3.4.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.2.3.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.2.3.5.1
First, use the positive value of the to find the first solution.
Step 1.2.3.5.2
Next, use the negative value of the to find the second solution.
Step 1.2.3.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.3
Evaluate when .
Step 1.3.1
Substitute for .
Step 1.3.2
Cancel the common factor of .
Step 1.3.2.1
Factor out of .
Step 1.3.2.2
Cancel the common factor.
Step 1.3.2.3
Rewrite the expression.
Step 1.4
Substitute for in and solve for .
Step 1.4.1
Multiply by .
Step 1.4.2
Multiply .
Step 1.4.2.1
Multiply by .
Step 1.4.2.2
Multiply by .
Step 1.5
Evaluate when .
Step 1.5.1
Substitute for .
Step 1.5.2
Simplify .
Step 1.5.2.1
Cancel the common factor of .
Step 1.5.2.1.1
Move the leading negative in into the numerator.
Step 1.5.2.1.2
Factor out of .
Step 1.5.2.1.3
Cancel the common factor.
Step 1.5.2.1.4
Rewrite the expression.
Step 1.5.2.2
Multiply by .
Step 1.6
Substitute for in and solve for .
Step 1.6.1
Remove parentheses.
Step 1.6.2
Simplify .
Step 1.6.2.1
Rewrite as .
Step 1.6.2.2
Multiply .
Step 1.6.2.2.1
Multiply by .
Step 1.6.2.2.2
Multiply by .
Step 1.7
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
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
Since is constant with respect to , move out of the integral.
Step 3.5
By the Power Rule, the integral of with respect to is .
Step 3.6
Combine and .
Step 3.7
Since is constant with respect to , move out of the integral.
Step 3.8
Since is constant with respect to , move out of the integral.
Step 3.9
Multiply by .
Step 3.10
The integral of with respect to is .
Step 3.11
Simplify the answer.
Step 3.11.1
Substitute and simplify.
Step 3.11.1.1
Evaluate at and at .
Step 3.11.1.2
Evaluate at and at .
Step 3.11.1.3
Simplify.
Step 3.11.1.3.1
Factor out of .
Step 3.11.1.3.2
Apply the product rule to .
Step 3.11.1.3.3
Raise to the power of .
Step 3.11.1.3.4
Multiply by .
Step 3.11.1.3.5
Combine the numerators over the common denominator.
Step 3.11.1.3.6
Simplify the numerator.
Step 3.11.1.3.6.1
Divide by .
Step 3.11.1.3.6.2
Raise to the power of .
Step 3.11.1.3.6.3
Divide by .
Step 3.11.1.3.6.4
Raise to the power of .
Step 3.11.1.3.6.5
Multiply by .
Step 3.11.1.3.6.6
Subtract from .
Step 3.11.1.3.7
Cancel the common factor of and .
Step 3.11.1.3.7.1
Factor out of .
Step 3.11.1.3.7.2
Cancel the common factors.
Step 3.11.1.3.7.2.1
Factor out of .
Step 3.11.1.3.7.2.2
Cancel the common factor.
Step 3.11.1.3.7.2.3
Rewrite the expression.
Step 3.11.1.3.7.2.4
Divide by .
Step 3.11.1.3.8
Multiply by .
Step 3.11.1.3.9
Subtract from .
Step 3.11.2
Use the quotient property of logarithms, .
Step 3.11.3
Simplify.
Step 3.11.3.1
is approximately which is positive so remove the absolute value
Step 3.11.3.2
is approximately which is negative so negate and remove the absolute value
Step 3.11.3.3
Cancel the common factor of .
Step 3.11.3.3.1
Cancel the common factor.
Step 3.11.3.3.2
Rewrite the expression.
Step 3.11.3.4
The natural logarithm of is .
Step 3.11.3.5
Multiply by .
Step 4