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Volume with known Cross-Section

Think about you are cutting a steak to slide. What is so fantastic is that each slice of steak represents a cross-sectional area of the whole steak, which means when you put them together, you will get the volume of the solid steak.

Cross Section is a topic that we learn in Calculus II; it is common in daily life. However, it is not easy to do it in Calculus for a beginner. So, in this following Blog, I will use a 3-D model and other visual ways to introduce how to calculate the volume of a 3-Dimensional solid by taking an area and building up from that area using known cross-sections.

How to do this? I will use a example from calculus book to explain this. The flat base of a solid sits in the xy-plane in the region bounded by the graphs of y=0 and \(y=e^{−x}\) between \(x=0\) and \(x=1\).Find volume of this solid if cross-sections are perpendicular to the x-axis and they are squares. The following graph shows that the bounded function in 2-D.
Firstly, we could use the knowledge we have to approximate it, such that we could cut the function to 10 pieces that have the same high, 0.1. In order to calculate those pieces volumes, we could use the cross-sectional area multiple high, such that \[V = h A(x) = h (e^{-x})^2 \] IIn order to help us to visualize and understand the approximation, I built a 3D model, and I also calculated the volume of each part and added them together to get the total volume.
Frome the following approximation we could use intergral to calculate it, which will be more accurate, such that \[V = \int^1_0 A(x) dx = \int^1_0 (e^{-x})^2 dx\ = \int^1_0e^{-2x}dx = -\frac{1}{2}\int^1_0 e^{-2x}d(-2x) = -\frac{1}{2}e^{-2x}\bigg|^1_0 = -\frac{1}{2}(e^{-2}-1) \approx 0.4323\]To From the equation above, we know that the accurate volume of this solid is 0.4323. We could compare with the approximate we did in table 0.477. They are really closed. If we cut the pieces with smaller and smaller high, we will infinitely close the accurate answer.

Author: Yueqi Li (Nicole)
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