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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
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 1.2.2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 1.2.3
Simplify each side of the equation.
Step 1.2.3.1
Use to rewrite as .
Step 1.2.3.2
Simplify the left side.
Step 1.2.3.2.1
Simplify .
Step 1.2.3.2.1.1
Apply the product rule to .
Step 1.2.3.2.1.2
Raise to the power of .
Step 1.2.3.2.1.3
Multiply the exponents in .
Step 1.2.3.2.1.3.1
Apply the power rule and multiply exponents, .
Step 1.2.3.2.1.3.2
Cancel the common factor of .
Step 1.2.3.2.1.3.2.1
Cancel the common factor.
Step 1.2.3.2.1.3.2.2
Rewrite the expression.
Step 1.2.3.2.1.4
Simplify.
Step 1.2.4
Solve for .
Step 1.2.4.1
Subtract from both sides of the equation.
Step 1.2.4.2
Factor the left side of the equation.
Step 1.2.4.2.1
Let . Substitute for all occurrences of .
Step 1.2.4.2.2
Factor out of .
Step 1.2.4.2.2.1
Factor out of .
Step 1.2.4.2.2.2
Factor out of .
Step 1.2.4.2.2.3
Factor out of .
Step 1.2.4.2.3
Replace all occurrences of with .
Step 1.2.4.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4.4
Set equal to .
Step 1.2.4.5
Set equal to and solve for .
Step 1.2.4.5.1
Set equal to .
Step 1.2.4.5.2
Solve for .
Step 1.2.4.5.2.1
Subtract from both sides of the equation.
Step 1.2.4.5.2.2
Divide each term in by and simplify.
Step 1.2.4.5.2.2.1
Divide each term in by .
Step 1.2.4.5.2.2.2
Simplify the left side.
Step 1.2.4.5.2.2.2.1
Dividing two negative values results in a positive value.
Step 1.2.4.5.2.2.2.2
Divide by .
Step 1.2.4.5.2.2.3
Simplify the right side.
Step 1.2.4.5.2.2.3.1
Divide by .
Step 1.2.4.6
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
Simplify .
Step 1.3.2.1
Remove parentheses.
Step 1.3.2.2
Rewrite as .
Step 1.3.2.3
Pull terms out from under the radical, assuming positive real numbers.
Step 1.3.2.4
Multiply by .
Step 1.4
Evaluate when .
Step 1.4.1
Substitute for .
Step 1.4.2
Simplify .
Step 1.4.2.1
Remove parentheses.
Step 1.4.2.2
Rewrite as .
Step 1.4.2.3
Pull terms out from under the radical, assuming positive real numbers.
Step 1.4.2.4
Multiply by .
Step 1.5
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
Use to rewrite as .
Step 3.6
By the Power Rule, the integral of with respect to is .
Step 3.7
Combine and .
Step 3.8
Since is constant with respect to , move out of the integral.
Step 3.9
By the Power Rule, the integral of with respect to is .
Step 3.10
Simplify the answer.
Step 3.10.1
Combine and .
Step 3.10.2
Substitute and simplify.
Step 3.10.2.1
Evaluate at and at .
Step 3.10.2.2
Evaluate at and at .
Step 3.10.2.3
Simplify.
Step 3.10.2.3.1
Rewrite as .
Step 3.10.2.3.2
Apply the power rule and multiply exponents, .
Step 3.10.2.3.3
Cancel the common factor of .
Step 3.10.2.3.3.1
Cancel the common factor.
Step 3.10.2.3.3.2
Rewrite the expression.
Step 3.10.2.3.4
Raise to the power of .
Step 3.10.2.3.5
Multiply by .
Step 3.10.2.3.6
Rewrite as .
Step 3.10.2.3.7
Apply the power rule and multiply exponents, .
Step 3.10.2.3.8
Cancel the common factor of .
Step 3.10.2.3.8.1
Cancel the common factor.
Step 3.10.2.3.8.2
Rewrite the expression.
Step 3.10.2.3.9
Raising to any positive power yields .
Step 3.10.2.3.10
Multiply by .
Step 3.10.2.3.11
Cancel the common factor of and .
Step 3.10.2.3.11.1
Factor out of .
Step 3.10.2.3.11.2
Cancel the common factors.
Step 3.10.2.3.11.2.1
Factor out of .
Step 3.10.2.3.11.2.2
Cancel the common factor.
Step 3.10.2.3.11.2.3
Rewrite the expression.
Step 3.10.2.3.11.2.4
Divide by .
Step 3.10.2.3.12
Multiply by .
Step 3.10.2.3.13
Add and .
Step 3.10.2.3.14
Combine and .
Step 3.10.2.3.15
Multiply by .
Step 3.10.2.3.16
Raise to the power of .
Step 3.10.2.3.17
Cancel the common factor of and .
Step 3.10.2.3.17.1
Factor out of .
Step 3.10.2.3.17.2
Cancel the common factors.
Step 3.10.2.3.17.2.1
Factor out of .
Step 3.10.2.3.17.2.2
Cancel the common factor.
Step 3.10.2.3.17.2.3
Rewrite the expression.
Step 3.10.2.3.17.2.4
Divide by .
Step 3.10.2.3.18
Raising to any positive power yields .
Step 3.10.2.3.19
Cancel the common factor of and .
Step 3.10.2.3.19.1
Factor out of .
Step 3.10.2.3.19.2
Cancel the common factors.
Step 3.10.2.3.19.2.1
Factor out of .
Step 3.10.2.3.19.2.2
Cancel the common factor.
Step 3.10.2.3.19.2.3
Rewrite the expression.
Step 3.10.2.3.19.2.4
Divide by .
Step 3.10.2.3.20
Multiply by .
Step 3.10.2.3.21
Add and .
Step 3.10.2.3.22
Multiply by .
Step 3.10.2.3.23
To write as a fraction with a common denominator, multiply by .
Step 3.10.2.3.24
Combine and .
Step 3.10.2.3.25
Combine the numerators over the common denominator.
Step 3.10.2.3.26
Simplify the numerator.
Step 3.10.2.3.26.1
Multiply by .
Step 3.10.2.3.26.2
Subtract from .
Step 4