Calculus: integration question

Question

In regards to the graph of f(x)=3x²ln(x), why does it lie on or above the x-axis for all values of x>=1 and what expression would give the area between this graph and the x-axis from say x=1 to x=4?

Answer 1

Question 1: Why does f(x)=3x²ln(x) lie on or above the x-axis for all values of x>=1?

Because the value of ln(x) is the sign changer and that determines whether f(x) is above the x-axis (positive), below the x-axis (negative) or on the x-axis (zero).

If 0 < x <1, then f(x) has negative value and that would be below the x-axis on the graph because ln (x) is negative.

If x <= 0, then f(x) is undefined.

However, if x=1, f(1) lies on the x-axis because f(1)=0. Only then ln(x) is positive or f(x) is above the x-axis when x>1.

Question 2: What expression would give the area between this graph and the x-axis from say x=1 to x=4?

In Calculus, we can express area of a region using definite integrals.

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 3x² ln(x) dx

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