1. Volume of Solids with Known Cross Sections
Honors Calculus
Keeper 36

2. Volumes of solids with known Cross sections
For cross sections of area 𝐴(𝑥) taken perpendicular to the 𝑥−𝑎𝑥𝑖𝑠, 𝑉𝑜𝑙𝑢𝑚𝑒= 𝑎 𝑏 𝐴 𝑥 𝑑𝑥 For cross sections of area 𝐴(𝑦) taken perpendicular to the 𝑦−𝑎𝑥𝑖𝑠 𝑉𝑜𝑙𝑢𝑚𝑒= 𝑎 𝑏 𝐴 𝑦 𝑑𝑦

3. Steps for Finding Volume
Draw the base on the 𝑥𝑦−plane
Draw a representative cross section
Find an area formula for the cross section 𝐴(𝑥) or 𝐴 𝑦
Set up integral with bounds
Integrate your area formula and use FTC on your bounds

4. Important Area formulas to know!
Square
Semicircle
Rectangle
Isosceles Right Triangle with base as leg
𝐴= 𝑏 2
𝐴= 1 2 𝜋 𝑏 2 2
𝐴=bh
𝑎= 1 2 𝑏 2

5. Example
Find the volume of the solid whose base is the region bounded by 𝑦= 𝑥 , 𝑥=9, and the 𝑥−𝑎𝑥𝑖𝑠. Bases of cross-sections are perpendicular to the 𝑥−𝑎𝑥𝑖𝑠. SQUARES:

6. Example
Find the volume of the solid whose base is the region bounded by 𝑦= 𝑥 , 𝑥=9, and the 𝑥−𝑎𝑥𝑖𝑠. Bases of cross-sections are perpendicular to the 𝑥−𝑎𝑥𝑖𝑠. SEMICIRCLES:

7. Example
Find the volume of the solid whose base is the region bounded by 𝑦= 𝑥 , 𝑥=9, and the 𝑥−𝑎𝑥𝑖𝑠. Bases of cross-sections are perpendicular to the 𝑥−𝑎𝑥𝑖𝑠. RECTANGLE WITH ℎ= 1 2 𝑏:

8. Example
Find the volume of the solid whose base is the region bounded by 𝑦= 𝑥 , 𝑥=9, and the 𝑥−𝑎𝑥𝑖𝑠. Bases of cross-sections are perpendicular to the 𝑥−𝑎𝑥𝑖𝑠. ISOSCELES RIGHT TRIANGLE WITH BASE AS LEG:

9. Example
Find the volume of the solid whose base is the region bounded by 𝑦= 𝑥 2 and 𝑦=9. Bases of cross- sections are perpendicular to the y−𝑎𝑥𝑖𝑠. SQUARES:

10. Example
Find the volume of the solid whose base is the region bounded by 𝑦= 𝑥 2 and 𝑦=9. Bases of cross- sections are perpendicular to the y−𝑎𝑥𝑖𝑠. SEMICIRCLES:

11. example
A solid has a circular base of radius 2 in the 𝑥𝑦- plane. Cross sections perpendicular to the x-axis are in the shape of isosceles right triangles with their hypotenuse as the base of the solid. Find the volume of the solid.

12. Example
Find the volume of the solid whose base is the region bounded between the curves 𝑦=𝑥 and 𝑦= 𝑥 2 , and whose cross sections perpendicular to the 𝑥−𝑎𝑥𝑖𝑠 are squares.

13. Example
The base of a certain solid is the region enclosed by 𝑦= 𝑥 , 𝑦=0, 𝑎𝑛𝑑 𝑥=4. Every cross section perpendicular to the x-axis is a semicircle with its diameter across the base. Find the volume of the solid.

14. Example
Find the volume of the solid whose base is the region bounded by 𝑓 𝑥 =1− 𝑥 2 , 𝑔 𝑥 =−1+ 𝑥 2 , 𝑎𝑛𝑑 𝑥=0. The cross sections are equilateral triangles.