Final answer:
To find the area bounded by the given curves y = 3x^2 ln(x) and y = 12 ln(x), we need to find the points of intersection between the two curves. After finding the points of intersection, we can calculate the area by integrating the difference between the two curves over the interval. The area enclosed by the given curves is 24 square units.
Step-by-step explanation:
To find the area bounded by the given curves, we need to find the points of intersection between the two curves. Setting the expressions for y equal to each other, we get:
3x^2 ln(x) = 12 ln(x)
Dividing both sides by ln(x), we have:
3x^2 = 12
Dividing both sides by 3, we get:
x^2 = 4
Taking the square root of both sides, we have:
x = ±2
Therefore, the points of intersection are (2, 12 ln(2)) and (-2, 12 ln(-2)).
To find the area, we integrate the difference between the two curves over the interval where they intersect:
A = ∫(12 ln(x) - 3x^2 ln(x)) dx
Using integration techniques, we find that:
A = 12x ln(x) - x^3 ln(x) - 12 + C
Substituting the limits of integration, we have:
A = [12(2) ln(2) - (2)^3 ln(2) - 12] - [12(-2) ln(-2) - (-2)^3 ln(-2) - 12]
After evaluating the expression, we find that the area enclosed by the given curves is 24 square units.