Calculating the Volume of a Solid with Square Cross-Sections

This article delves into the method of calculating the volume of a solid whose base is a region in the first quadrant bounded by the y-axis, x-axis, the graph of y e^x, and the vertical line x 1. This special solid has square cross-sections perpendicular to the x-axis. We will use integration to find the volume step by step.

Understanding the Problem

The base of the solid is defined by the region in the first quadrant, bounded by the x-axis, the y-axis, the curve y e^x, and the vertical line x 1. For each vertical cross-section of this solid at a distance x from the y-axis, the cross-section is a square with a side length equal to the height y e^x.

The area of each square cross-section is therefore e^{2x}. To find the volume of the solid, we will integrate the area of these cross-sections over the given interval.

Step-by-Step Calculation

Let's begin by setting up the integral that will give us the volume.

Area of a Square Cross-Section

The area of a square cross-section at a distance x from the y-axis is:

Area (e^x)^2 e^{2x}

Setting Up the Integral

To find the volume of the solid, we will integrate the area of the square cross-sections from x 0 to x 1 over the x-axis. The integral is:

V int_0^1 e^{2x} , dx

Evaluating the Integral

To evaluate this integral, we use the substitution method. Let:

u 2x du 2 , dx frac{du}{2} dx

Substituting these into the integral, we get:

V frac{1}{2} int_0^2 e^u , du

Now, we can integrate:

V frac{1}{2} left[ e^u right]_0^2 frac{1}{2} left( e^2 - e^0 right)

Simplifying this, we find:

V frac{1}{2} left( e^2 - 1 right) V approx frac{1}{2} (7.389 - 1) approx frac{1}{2} times 6.389 approx 3.1945 text{units}^3

Summary

The volume of the solid is given by the integral of the square cross-sections from x 0 to x 1. The final volume is approximately 3.195 text{cubic units}.

Conclusion

Using the method of integration, we have calculated the volume of a solid with a specific base, where each cross-section is a square. The process involves understanding the geometric properties of the solid and utilizing calculus to solve the problem accurately.