Enter a problem...
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
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
Subtract from both sides of the equation.
Step 1.2.3
Factor .
Step 1.2.3.1
Use to rewrite as .
Step 1.2.3.2
Factor out of .
Step 1.2.3.2.1
Multiply by .
Step 1.2.3.2.2
Factor out of .
Step 1.2.3.2.3
Factor out of .
Step 1.2.4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.5
Set equal to and solve for .
Step 1.2.5.1
Set equal to .
Step 1.2.5.2
Solve for .
Step 1.2.5.2.1
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 1.2.5.2.2
Simplify the exponent.
Step 1.2.5.2.2.1
Simplify the left side.
Step 1.2.5.2.2.1.1
Simplify .
Step 1.2.5.2.2.1.1.1
Multiply the exponents in .
Step 1.2.5.2.2.1.1.1.1
Apply the power rule and multiply exponents, .
Step 1.2.5.2.2.1.1.1.2
Cancel the common factor of .
Step 1.2.5.2.2.1.1.1.2.1
Cancel the common factor.
Step 1.2.5.2.2.1.1.1.2.2
Rewrite the expression.
Step 1.2.5.2.2.1.1.2
Simplify.
Step 1.2.5.2.2.2
Simplify the right side.
Step 1.2.5.2.2.2.1
Raising to any positive power yields .
Step 1.2.6
Set equal to and solve for .
Step 1.2.6.1
Set equal to .
Step 1.2.6.2
Solve for .
Step 1.2.6.2.1
Subtract from both sides of the equation.
Step 1.2.6.2.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 1.2.6.2.3
Simplify the exponent.
Step 1.2.6.2.3.1
Simplify the left side.
Step 1.2.6.2.3.1.1
Simplify .
Step 1.2.6.2.3.1.1.1
Simplify the expression.
Step 1.2.6.2.3.1.1.1.1
Apply the product rule to .
Step 1.2.6.2.3.1.1.1.2
Rewrite as .
Step 1.2.6.2.3.1.1.1.3
Apply the power rule and multiply exponents, .
Step 1.2.6.2.3.1.1.2
Cancel the common factor of .
Step 1.2.6.2.3.1.1.2.1
Cancel the common factor.
Step 1.2.6.2.3.1.1.2.2
Rewrite the expression.
Step 1.2.6.2.3.1.1.3
Simplify the expression.
Step 1.2.6.2.3.1.1.3.1
Raise to the power of .
Step 1.2.6.2.3.1.1.3.2
Multiply by .
Step 1.2.6.2.3.1.1.3.3
Multiply the exponents in .
Step 1.2.6.2.3.1.1.3.3.1
Apply the power rule and multiply exponents, .
Step 1.2.6.2.3.1.1.3.3.2
Cancel the common factor of .
Step 1.2.6.2.3.1.1.3.3.2.1
Cancel the common factor.
Step 1.2.6.2.3.1.1.3.3.2.2
Rewrite the expression.
Step 1.2.6.2.3.1.1.3.3.3
Cancel the common factor of .
Step 1.2.6.2.3.1.1.3.3.3.1
Cancel the common factor.
Step 1.2.6.2.3.1.1.3.3.3.2
Rewrite the expression.
Step 1.2.6.2.3.1.1.4
Simplify.
Step 1.2.6.2.3.2
Simplify the right side.
Step 1.2.6.2.3.2.1
Simplify .
Step 1.2.6.2.3.2.1.1
Simplify the expression.
Step 1.2.6.2.3.2.1.1.1
Rewrite as .
Step 1.2.6.2.3.2.1.1.2
Apply the power rule and multiply exponents, .
Step 1.2.6.2.3.2.1.2
Cancel the common factor of .
Step 1.2.6.2.3.2.1.2.1
Cancel the common factor.
Step 1.2.6.2.3.2.1.2.2
Rewrite the expression.
Step 1.2.6.2.3.2.1.3
Raise to the power of .
Step 1.2.7
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 1.4
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
Any root of is .
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
Use to rewrite as .
Step 3.5
By the Power Rule, the integral of with respect to is .
Step 3.6
Since is constant with respect to , move out of the integral.
Step 3.7
By the Power Rule, the integral of with respect to is .
Step 3.8
Simplify the answer.
Step 3.8.1
Combine and .
Step 3.8.2
Substitute and simplify.
Step 3.8.2.1
Evaluate at and at .
Step 3.8.2.2
Evaluate at and at .
Step 3.8.2.3
Simplify.
Step 3.8.2.3.1
One to any power is one.
Step 3.8.2.3.2
Multiply by .
Step 3.8.2.3.3
Rewrite as .
Step 3.8.2.3.4
Apply the power rule and multiply exponents, .
Step 3.8.2.3.5
Cancel the common factor of .
Step 3.8.2.3.5.1
Cancel the common factor.
Step 3.8.2.3.5.2
Rewrite the expression.
Step 3.8.2.3.6
Raising to any positive power yields .
Step 3.8.2.3.7
Multiply by .
Step 3.8.2.3.8
Multiply by .
Step 3.8.2.3.9
Add and .
Step 3.8.2.3.10
One to any power is one.
Step 3.8.2.3.11
Raising to any positive power yields .
Step 3.8.2.3.12
Cancel the common factor of and .
Step 3.8.2.3.12.1
Factor out of .
Step 3.8.2.3.12.2
Cancel the common factors.
Step 3.8.2.3.12.2.1
Factor out of .
Step 3.8.2.3.12.2.2
Cancel the common factor.
Step 3.8.2.3.12.2.3
Rewrite the expression.
Step 3.8.2.3.12.2.4
Divide by .
Step 3.8.2.3.13
Multiply by .
Step 3.8.2.3.14
Add and .
Step 3.8.2.3.15
To write as a fraction with a common denominator, multiply by .
Step 3.8.2.3.16
To write as a fraction with a common denominator, multiply by .
Step 3.8.2.3.17
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 3.8.2.3.17.1
Multiply by .
Step 3.8.2.3.17.2
Multiply by .
Step 3.8.2.3.17.3
Multiply by .
Step 3.8.2.3.17.4
Multiply by .
Step 3.8.2.3.18
Combine the numerators over the common denominator.
Step 3.8.2.3.19
Simplify the numerator.
Step 3.8.2.3.19.1
Multiply by .
Step 3.8.2.3.19.2
Subtract from .
Step 4