# Calculus Examples

,
Solve by substitution to find the intersection between the curves.
Substitute for into then solve for .
Replace with in the equation.
Solve the equation for .
Move all terms containing to the left side of the equation.
Subtract from both sides of the equation.
Combine the opposite terms in .
Subtract from .
Take the root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Substitute for into then solve for .
Replace with in the equation.
Remove the parentheses around the expression .
Substitute for into then solve for .
Replace with in the equation.
Remove the parentheses around the expression .
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 .
Combine the integrals into a single integral.
Apply the distributive property.
Combine the opposite terms in .
Subtract from .
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
Combine and .
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Simplify.
Multiply by .
Rewrite as .
Raise to the power of .
Factor out of .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Move the negative in front of the fraction.
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Simplify.
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Multiply by .
Subtract from .
The result can be shown in multiple forms.
Exact Form:
Decimal Form: