What is the net area between #f(x)=ln(x^3-x+2)/x^2# in #x in[1,2] # and the x-axis?

Question

What is the net area  between #f(x)=ln(x^3-x+2)/x^2# in #x  in[1,2]    # and the x-axis?

Isaiah Allen · Accepted Answer

Area #~~0.601# The net area of a function #f# in the interval #[a,b]# is given by #A=int_a^bf(x)dx#. #A=int_1^2ln(x^3-x+3)/x^2dx# From wolfram alpha, we have #A~~0.601#

Elijah Abram · Answer

undefined To find the net area between the curve $ f(x) = \frac{\ln(x^3 - x + 2)}{x^2} $ and the x-axis on the interval $[1, 2]$, we need to integrate the absolute value of the function from $ x = 1 $ to $ x = 2 $:

$$
\text{Net area} = \int_{1}^{2} |f(x)| \, dx
$$

First, determine the critical points by setting the function equal to zero:

$$
x^3 - x + 2 = 0
$$

Using numerical methods, solve for $ x $ to find the critical point within the interval $[1, 2]$.

Next, evaluate the function $ f(x) $ at the critical point and the endpoints of the interval to determine the sign of the function within the interval.

Then, split the integral into segments based on the sign changes of $ f(x) $ within the interval.

Finally, integrate each segment separately and sum the absolute values of the results to find the net area between the curve and the x-axis within the given interval.