# Find the Area Between the Curves y=3x^2 natural log of x , y=12 natural log of x

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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 .
Simplify the left side of the equation.
Remove parentheses.
Simplify by moving inside the logarithm.
Simplify by moving inside the logarithm.
Graph each side of the equation. The solution is the x-value of the point of intersection.
Substitute for into then solve for .
Replace with in the equation.
Simplify .
The natural logarithm of is .
Multiply by .
Substitute for into then solve for .
Replace with in the equation.
Simplify .
Simplify by moving inside the logarithm.
Raise to the power of .
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.
Simplify each term.
Simplify by moving inside the logarithm.
Simplify by moving inside the logarithm.
Rewrite as .
Simplify each term.
Rewrite using the commutative property of multiplication.
Simplify by moving inside the logarithm.
Simplify by moving inside the logarithm.
Find the Area Between the Curves y=3x^2 natural log of x , y=12 natural log of x