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Unit 8

8.8 Volumes with Cross Sections: Triangles and Semicircles

1 min readโ€ขjune 8, 2020

anusha

Anusha Tekumulla


AP Calculus AB/BCย โ™พ๏ธ

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Using Triangles as Cross Sections

This topic is very similar to topic 8.7 except now weโ€™re using triangles and semicircles as cross sections. To find the area of a shape using a triangle or semicircle cross section, take a look at the example below.ย 

๐Ÿ” Example Problem: Finding the Volumes of a Shape with Triangle Cross Sectionsย ย 

Letโ€™s say we are asked to find the volume of a solid whose base is the circle x^2 + y^2 = 4, where the cross sections perpendicular to the x-axis are all equilateral triangles.ย NOTE: Sometimes you'll be given isosceles right triangles or semicircles. Make sure you using the right formula!
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(663).png?alt=media&token=c59f027d-ffdd-4a51-8445-9a37bea7e8ec
We find the side of the triangle by doing top minus bottom. Thus, the side of the triangle is 2โˆš4 - x^2. Now, we use the formula for the equilateral triangle to find the area of the cross section (A = (side)2 *ย  โˆš3 / 4). We can now find the volume by integrating the area function from x = -2 to x = 2.ย 

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