The volume of the solid formed by rotating the region bounded by $$y = \ln(x)$$ and the x-axis from $$x = 1$$ to $$x = e$$ about the x-axis can be expressed as an integral. Which of the following integrals correctly represents this volume?

Bounded Regions from Intersecting Curves

Volume With Square and Rectangular Cross-Sections

Volume of a Solid of Revolution

Volume with Known Cross-Sections

Solve for x: $$\frac{x}{\sqrt{x+5}}-\frac{\sqrt{x}}{x+1}=0$$.

Exponential Growth Differential Equation

Maximum Rate of Logistic Growth

Separation of Variables Method

Logistic Growth Rate Near Carrying Capacity

Elementary Antiderivatives of Rational Functions

Interpreting the Integral of Acceleration

Riemann Sum Definition of an Integral

Accuracy of Riemann Sum Methods

Improper Integral Convergence

Concavity and the Second Derivative

Consider the function $$h(x)= x^3 - 3*x + 1$$ with derivative $$h'(x)= 3*(x - 1)*(x + 1)$$. This function has a relative maximum at x = -1 and a relative minimum at x = 1. Also, h is increasing for x < -1 and decreasing on the interval (-1, 1). Rank the following behaviors in increasing order of x-values: 'Increasing on $$(-\infty, -1)$$', 'Relative maximum at x = -1', 'Decreasing on $$(-1, 1)$$', and 'Relative minimum at x = 1'.

Product and Chain Rule Differentiation

Points of Inflection and the Second Derivative

Formulating an Area Optimization Function