# Find the Area Between the Curves x=7-7y^2 , x=7y^2-7 ,
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.
Add to both sides of the equation.
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Any root of is .
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.
Simplify .
Simplify each term.
One to any power is one.
Multiply by .
Subtract from .
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.
Simplify each term.
Apply the distributive property.
Multiply by .
Multiply by .
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.
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Simplify.
One to any power is one.
Raise to the power of .
Move the negative in front of the fraction.
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Combine and .
Multiply by .
Move the negative in front of the fraction.
Multiply by .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .     