# Calculus Examples

,
Solve by substitution to find the intersection between the curves.
Eliminate the equal sides of each equation and combine.
Solve for .
Move all terms containing to the left side of the equation.
Subtract from both sides of the equation.
Subtract from .
Factor the left side of the equation.
Let . Substitute for all occurrences of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Replace all occurrences of with .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to .
Set equal to and solve for .
Set equal to .
Solve for .
Subtract from both sides of the equation.
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify the right side.
Dividing two negative values results in a positive value.
The final solution is all the values that make true.
Evaluate when .
Substitute for .
Substitute for in and solve for .
Remove parentheses.
Raising to any positive power yields .
Evaluate when .
Substitute for .
Simplify .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
The solution to the system is the complete set of ordered pairs that are valid solutions.
Reorder and .
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.
Subtract from .
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
Combine and .
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.
Raising to any positive power yields .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Raising to any positive power yields .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Simplify.
Simplify each term.
Simplify the numerator.
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify the numerator.
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply by .
Multiply by .
Multiply .
Combine and .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .