Imagine you sliced up a watermelon … yum!
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
What is so cool is that each slice represents a wedge or cross-sectional area of the entire watermelon. And this means that when you put all of the watermelon slices together, they create the volume of the solid.
So a cross-section is a shape that would be exposed by making a straight cut or slice through something. And we can find the volume by cross sections for all types of solids. Awesome!
So first, we will quickly review how to find the area between two curves because area and volume are intrinsically linked.
Why? Because the antiderivative of area is volume!
Here’ another way to think about this… we know that derivatives “go down” in the sense of decreasing exponents as we go from position to velocity to acceleration, etc. And integration “goes up” and takes a derivative and then finds the position.
The same thing with finding volume; integrate the area, and we find the volume of the solid on the closed interval!
Then we will learn how to create a 3-Dimensional solid by taking an area and building up from that area using known cross-sections.
How do we do this?
We slice the region perpendicular to either the x-axis or the y-axis, as Cliff Notes accurately states, and then we use a definite integral to find the volume.
Formulas for Known Cross Sections
Finally, we will learn the five necessary forms for finding volume using cross-sections (i.e., squares, equilateral triangles, isosceles triangles, right triangles, semicircles, and rectangles), and learn how to apply them to all different types of questions.
Volumes with Known Cross Sections Video
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