1 min readโขjune 8, 2020

Anusha Tekumulla

This topic is very similar to topic 8.7 except now weโre **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

Letโs 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!

We find the side of the triangle by doing **top minus bottom**. Thus, the side of the triangle is 2โ4 - x^2. Now, we **use the formula for the equilateral triangle to find the area of the cross section** (A = (side)2 *ย โ3 / 4). We can now find the volume by **integrating the area function** from x = -2 to x = 2.ย

Browse Study Guides By Unit

๐Unit 1 โ Limits & Continuity

๐คUnit 2 โ Fundamentals of Differentiation

๐ค๐ฝUnit 3 โ Composite, Implicit, & Inverse Functions

๐Unit 4 โ Contextual Applications of Differentiation

โจUnit 5 โ Analytical Applications of Differentiation

๐ฅUnit 6 โ Integration & Accumulation of Change

๐Unit 7 โ Differential Equations

๐ถUnit 8 โ Applications of Integration

๐ฆUnit 9 โ Parametric Equations, Polar Coordinates, & Vector-Valued Functions (BC Only)

โพUnit 10 โ Infinite Sequences & Series (BC Only)

๐งMultiple Choice Questions (MCQ)

โ๏ธFree Response Questions (FRQ)

๐Big Reviews: Finals & Exam Prep

ยฉ 2023 Fiveable Inc. All rights reserved.