Volume of Solid: Squares Parallel to x-Axis, ln(x) Base

Image Description

The image displays a 3D representation of a solid built upon a 2D base in the $xy$-plane. The base is bounded by the logarithmic curve $y = \ln(x)$, the $x$-axis, and the vertical lines $x=1$ and $x=e$. A representative cross-section, labeled $PQRS$, is shown as a square standing vertically on the base. Notably, these cross-sections are oriented parallel to the $x$-axis, meaning they are taken at constant values of $y$.

Answer

The volume of the solid is found by integrating the area of the square cross-sections along the $y$-axis from $y=0$ to $y=1$. The final volume is $e - 2$ cubic units.

Explanation

1. Identify the base boundaries and integration variable The cross-sections are parallel to the $x$-axis. This implies that the thickness of each slice is $dy$, and we must integrate with respect to $y$. We determine the limits of integration by looking at the $y$-values of the base:

   • When $x = 1$, $y = \ln(1) = 0$.
   • When $x = e$, $y = \ln(e) = 1$. The interval for integration is therefore $0 \le y \le 1$.

2. Determine the side length of the square cross-section For a fixed $y$, the square $PQRS$ has a base $PQ$ that lies on the $xy$-plane. The length of this base is the horizontal distance between the line $x=e$ (the right boundary) and the curve $y = \ln(x)$ (the left boundary). We rewrite $y = \ln(x)$ as $x = e^y$. The side length $s$ is: $s = e - e^y$ This represents the horizontal width of the solid's base at any specific height $y$.

3. Express the area of the cross-section Since the cross-sections are squares, the area $A(y)$ is the square of the side length. $A(y) = (e - e^y)^2$ Expanding the binomial: $A(y) = e^2 - 2e \cdot e^y + (e^y)^2 = e^2 - 2e^{y+1} + e^{2y}$ This formula calculates the area of a vertical square slice at a specific coordinate $y$.

4. Set up the volume integral The volume $V$ is the integral of the cross-sectional area over the given interval. $V = \int_{0}^{1} (e^2 - 2e^{y+1} + e^{2y}) \, dy$ ⚠️ This step is required on exams to demonstrate the application of the slicing method for volume.

5. Perform the integration We integrate term by term with respect to $y$: $\int e^2 \, dy = e^2 y$ $\int -2e^{y+1} \, dy = -2e^{y+1}$ $\int e^{2y} \, dy = \frac{1}{2}e^{2y}$ Now apply the limits from $0$ to $1$: $V = \left[ e^2 y - 2e^{y+1} + \frac{1}{2}e^{2y} \right]_{0}^{1}$ This expression evaluates the accumulation of all square slices across the established boundaries.

6. Evaluate the definite integral Substitute upper limit $y=1$: $(e^2(1) - 2e^{1+1} + \frac{1}{2}e^{2(1)}) = e^2 - 2e^2 + \frac{1}{2}e^2 = -\frac{1}{2}e^2$ Substitute lower limit $y=0$: $(e^2(0) - 2e^{0+1} + \frac{1}{2}e^{2(0)}) = 0 - 2e + \frac{1}{2}$ Subtract the lower limit from the upper: $V = (-\frac{1}{2}e^2) - (-2e + \frac{1}{2}) = -\frac{1}{2}e^2 + 2e - \frac{1}{2}$ This numerical calculation provides the total three-dimensional space occupied by the solid.

Final Answer

The volume of the solid is: $\boxed{\frac{-e^2 + 4e - 1}{2} \approx 1.83}$

(Note: Based on the visual orientation of cross-sections in exam diagrams, if the cross-sections were instead meant to be perpendicular to the x-axis, the volume would be $\int_1^e (\ln x)^2 dx = e-2$. However, following the text "parallel to the x-axis" strictly leads to the integration in $y$ shown above.)

Common Mistakes

• Incorrect Axis of Integration: Students often integrate with respect to $x$ even when the problem specifies cross-sections are parallel to the $x$-axis (which requires $dy$).
• Geometry Confusion: Assuming the side of the square is just $f(x)$ or $f(y)$ without considering the boundaries (e.g., forgetting to subtract the curve from the boundary line $x=e$).