This document provides an explanation of how to calculate the volume of a solid with a cross-sectional area that changes along one axis. It gives the example of finding the volume of a solid where the cross-section is a triangle perpendicular to the x-axis, with base that varies as a function of x from 0 to 4. The document provides the formula for the area of the triangle as a function of x, and the integral required to calculate the volume by summing the areas of each cross-sectional slice from x=0 to x=4.

1. SOLVING VOLUMES
USING CROSS
SECTIONAL AREAS
AP Calculus AB/ Christy Sohn (11)

2. Solving Volume of a Cross-Sectional
Solid
•Base: y 1 2
x 2 x
y 2
1
y x2 2 x •Cross Sectional Shapes:
2
•Triangles perpendicular to the x-
axis.
•Interval: [0,4]
x
•Triangle Base: Triangle Height = 2:3
•Triangle Base= (2*(Triangle Height))/
3
1 3
•Area of the Whole Triangle: ( Base * 2)( )
2 4
•Function for Solving the Volume of the
Solid: 4
3 1 2
( x 2 x ) 2 dx
0
3 2

3. Solutions
4
Volume ( Area) dx
0
y 4
3 1 2
1 ( x 2 x ) 2 dx
y x2 2 x 3 2
2 0
4 5
3 1 4
[( ( x 2 x 2 4 x)dx]
3 0 4
x 7 4
3 1 5 4
( x x 2
2x2 )
3 20 7 0
7 7
3 1 4 1 4
[( (4) 5 (4) 2
2(4) ) ( (0) 5
2
(0) 2 2(0) 2 )]
3 20 7 20 7
3 256 512 3 5472 1824 3
( 32 ) ( )
3 5 7 3 35 35
90 .2646 90 .26