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

$y = 3 x^{2} \ln (x)$ , $y = 12 \ln (x)$

Solve by substitution to find the intersection between the curves.

Tap for more steps…

Substitute $3 x^{2} \ln (x)$ for $y$ into $y = 12 \ln (x)$ then solve for $x$ .

Tap for more steps…

Replace $y$ with $3 x^{2} \ln (x)$ in the equation.

$3 x^{2} \ln (x) = 12 \ln (x)$

Solve the equation for $x$ .

Tap for more steps…

Simplify the left side of the equation.

Tap for more steps…

Remove parentheses.

$3 x^{2} \ln (x) = 12 \ln (x)$

Simplify $3 \ln (x)$ by moving $3$ inside the logarithm.

$x^{2} \ln (x^{3}) = 12 \ln (x)$

Simplify $12 \ln (x)$ by moving $12$ inside the logarithm.

$x^{2} \ln (x^{3}) = \ln (x^{12})$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 1, 2$

Substitute $1$ for $x$ into $y = 12 \ln (x)$ then solve for $y$ .

Tap for more steps…

Replace $x$ with $1$ in the equation.

$x = 1$

$y = 12 \ln (1)$

Simplify $12 \ln (1)$ .

Tap for more steps…

The natural logarithm of $1$ is $0$ .

$y = 12 \cdot 0$

Multiply $12$ by $0$ .

$y = 0$

Substitute $2$ for $x$ into $y = 12 \ln (x)$ then solve for $y$ .

Tap for more steps…

Replace $x$ with $2$ in the equation.

$x = 2$

$y = 12 \ln (2)$

Simplify $12 \ln (2)$ .

Tap for more steps…

Simplify $12 \ln (2)$ by moving $12$ inside the logarithm.

$y = \ln (2^{12})$

Raise $2$ to the power of $12$ .

$y = \ln (4096)$

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(1, 0)$

$(2, \ln (4096))$

$(1, 0)$

$(2, \ln (4096))$

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.

$A r e a = \int_{1}^{2} 3 x^{2} \ln (x) d x - \int_{1}^{2} 12 \ln (x) d x$

Integrate to find the area between $1$ and $2$ .

Tap for more steps…

Combine the integrals into a single integral.

$\int_{1}^{2} 3 x^{2} \ln (x) - (12 \ln (x)) d x$

Simplify each term.

Tap for more steps…

Simplify $3 \ln (x)$ by moving $3$ inside the logarithm.

$x^{2} \ln (x^{3}) - (12 \ln (x))$

Simplify $12 \ln (x)$ by moving $12$ inside the logarithm.

$x^{2} \ln (x^{3}) - \ln (x^{12})$

$\int_{1}^{2} x^{2} \ln (x^{3}) - \ln (x^{12}) d x$

Rewrite as $\int x^{2} (3 \ln (x)) - (12 \ln (x)) d x$ .

$\int_{1}^{2} x^{2} (3 \ln (x)) - (12 \ln (x)) d x$

Simplify each term.

Tap for more steps…

Rewrite using the commutative property of multiplication.

$\int_{1}^{2} 3 x^{2} \ln (x) - (12 \ln (x)) d x$

Simplify $3 \ln (x)$ by moving $3$ inside the logarithm.

$\int_{1}^{2} x^{2} \ln (x^{3}) - (12 \ln (x)) d x$

Simplify $12 \ln (x)$ by moving $12$ inside the logarithm.

$\int_{1}^{2} x^{2} \ln (x^{3}) - \ln (x^{12}) d x$

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

Download our
App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our App from the store

Create a High Performed UI/UX Design from a Silicon Valley.

Download our
App from the store