# Find the Area Between the Curves y = square root of x , y=1/2x , x=9

, ,
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 .
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify each side of the equation.
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Simplify.
Combine and .
Apply the product rule to .
Raise to the power of .
Solve for .
Subtract from both sides of the equation.
Multiply through by the least common denominator , then simplify.
Apply the distributive property.
Cancel the common factor of .
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Reorder and .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Simplify .
Change the to .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Simplify .
Change the to .
Subtract from .
The final answer is the combination of both solutions.
Substitute for into then solve for .
Replace with in the equation.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Substitute for into then solve for .
Replace with in the equation.
Multiply by .
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.
Combine and .
Split the single integral into multiple integrals.
Use to rewrite as .
By the Power Rule, the integral of with respect to is .
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 .
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Simplify.
Rewrite as .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Raise to the power of .
Combine and .
Multiply by .
Rewrite as .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Raising to any positive power yields .
Multiply by .
Multiply by .
Raise to the power of .
Combine and .
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 .
Raising to any positive power yields .
Multiply by .
Multiply by .
Multiply by .
Combine and .
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 .
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 .
Subtract from .
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.
Combine and .
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 .
Since is constant with respect to , move out of the integral.
Use to rewrite as .
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.
Raise to the power of .
Combine and .
Raise to the power of .
Multiply by .
Combine and .
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 .
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 .
Subtract from .
Multiply and .
Multiply by .
Rewrite as .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Raise to the power of .
Multiply by .
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 .
Rewrite as .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Raise to the power of .
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 .
Subtract from .
To write as a fraction with a common denominator, 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 and .
Multiply by .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Multiply by .
Subtract from .
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 and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Find the Area Between the Curves y = square root of x , y=1/2x , x=9

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