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  3. ZuAI AP College Board Calculus AB/BC
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Volumes with Cross Sections: Triangles and Semicircles

Benjamin Wright

Benjamin Wright

8 min read

Next Topic - Volume with Disc Method: Revolving Around the x- or y-Axis
Study Guide Overview

This study guide covers calculating volumes of solids with known cross-sections using integration. It focuses on cross-sections that are equilateral triangles, right isosceles triangles, and semicircles. The guide explains how to determine the area function A(x) for each shape and how to find the integration bounds. Example problems demonstrate setting up and solving the volume integral using these techniques. The general formula V = ∫ A(x) dx is emphasized throughout.

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Previous Topic - Volumes with Cross Sections: Squares and RectanglesNext Topic - Volume with Disc Method: Revolving Around the x- or y-Axis

Question 1 of 11

What is the general formula used to find the volume of a solid with known cross-sectional area A(x)A(x)A(x) from x=ax=ax=a to x=bx=bx=b?

V=A(x)⋅(b−a)V = A(x) \cdot (b-a)V=A(x)⋅(b−a)

V=∫abA(x)dxV = \int_a^b A(x) dxV=∫ab​A(x)dx

V=A(a)+A(b)2⋅(b−a)V = \frac{A(a)+A(b)}{2} \cdot (b-a)V=2A(a)+A(b)​⋅(b−a)

V=∫A(x)dxV = \int A(x) dxV=∫A(x)dx