1. Volume by Cross Sections
Section 7.2A
Calculus AP/Dual, Revised ©2017
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7.2A: Volume using Cross Sections

2. 7.2A: Volume using Cross Sections
Review of Area
How does one find area of a square?
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7.2A: Volume using Cross Sections

3. 7.2A: Volume using Cross Sections
Review of Area
How does one find area of a semi-circle?
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7.2A: Volume using Cross Sections

4. 7.2A: Volume using Cross Sections
Review of Area
How does one find area of a semi-circle? (or think of it this way)
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7.2A: Volume using Cross Sections

5. 7.2A: Volume using Cross Sections
Review of Area
How does one find area of an isosceles triangle?
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7.2A: Volume using Cross Sections

6. 7.2A: Volume using Cross Sections
Review of Area
How does one find area of an equilateral triangle?
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7.2A: Volume using Cross Sections

7. 7.2A: Volume using Cross Sections
Calculus in Motion
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7.2A: Volume using Cross Sections

8. 7.2A: Volume using Cross Sections
Slicing it Up
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7.2A: Volume using Cross Sections

9. 7.2A: Volume using Cross Sections
Slicing it Up
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7.2A: Volume using Cross Sections

10. 7.2A: Volume using Cross Sections
Equation
𝑨 𝒙 is the area of the cross sections (2 dimensions)
𝒅𝒙 is the with in respects to the axis
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7.2A: Volume using Cross Sections

11. 7.2A: Volume using Cross Sections
Cross Sections Method
For cross sections of area 𝑨 𝒙 taken perpendicular to the 𝒙-axis: 𝑽= 𝒂 𝒃 𝑨 𝒙 𝒅𝒙
For cross sections of area 𝑨 𝒚 taken perpendicular to the 𝒚-axis: 𝑽= 𝒄 𝒅 𝑨 𝒚 𝒅𝒚
Area Formulas:
Square: 𝑨= 𝒔 𝟐
Semicircle: 𝑨= 𝝅 𝒅 𝟐 𝟖
Equilateral Right Triangle: 𝑨= 𝒔 𝟐 𝟑 𝟒
Isosceles Right Triangle: 𝑨= 𝒔 𝟐 𝟐
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7.2A: Volume using Cross Sections

12. 7.2A: Volume using Cross Sections
4 Corner Example
Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are ______________. What is the volume of the solid? Squares | Semi Circles | Isosceles Right Triangle | Equilateral Right Triangle
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7.2A: Volume using Cross Sections

13. 7.2A: Volume using Cross Sections
Intersection Point
Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are ______________. What is the volume of the solid?
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7.2A: Volume using Cross Sections

14. 7.2A: Volume using Cross Sections
4 Corner Model – Squares
Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are SQUARES. What is the volume of the solid?
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7.2A: Volume using Cross Sections

15. 4 Corner Model – Semi Circles
Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are SEMICIRCLES. What is the volume of the solid?
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7.2A: Volume using Cross Sections

16. 4 Corner Model – Isosceles Right Triangles
Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are Isosceles Right Triangles. What is the volume?
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7.2A: Volume using Cross Sections

17. 4 Corner Model – Equilateral Right Triangles
Find the volume of a solid whose base is bounded by 𝒇 𝒙 =𝒙𝟐 and 𝒈 𝒙 =𝟒𝒙−𝒙𝟐 and which the cross sections are perpendicular to the 𝒙-axis are Equilateral Right Triangles. What is the volume?
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7.2A: Volume using Cross Sections

18. 7.2A: Volume using Cross Sections
Steps
Find the intersection points
Determine the cross section and what axis is it perpendicular to
Apply the PROPER cross section
Evaluate
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7.2A: Volume using Cross Sections

19. 7.2A: Volume using Cross Sections
Example 1 (with Calc)
Find the volume of the solid whose base is the region bounded by the graphs of 𝒇(𝒙)= 𝒙  and 𝒈(𝒙)= 𝒙 𝟐 𝟖 , if slices perpendicular to the base along the x-axis have cross sections that are squares.
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7.2A: Volume using Cross Sections

20. 7.2A: Volume using Cross Sections
Example 1 (with Calc)
Find the volume of the solid whose base is the region bounded by the graphs of 𝒇(𝒙)= 𝒙  and 𝒈(𝒙)= 𝒙 𝟐 𝟖 , if slices perpendicular to the base along the x-axis have cross sections that are squares.
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7.2A: Volume using Cross Sections

21. 7.2A: Volume using Cross Sections
Example 2 (Non-Calc)
The base is the triangle enclosed by 𝒙+𝒚=𝟒, the 𝒙-axis, and 𝒚-axis. The slices are made perpendicular to the 𝒚-axis are semicircles. Find the volume of the figure.
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7.2A: Volume using Cross Sections

22. 7.2A: Volume using Cross Sections
Example 2 (Non-Calc)
The base is the triangle enclosed by 𝒙+𝒚=𝟒, the 𝒙-axis, and 𝒚-axis. The slices are made perpendicular to the 𝒚-axis are semicircles. Find the volume of the figure.
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7.2A: Volume using Cross Sections

23. 7.2A: Volume using Cross Sections
Example 3
Find the volume of a solid whose base is bounded by the graphs 𝒚=𝒙 𝟑 , 𝒚=𝟎, and 𝒙=𝟏. Cross sections are perpendicular to the 𝒚-axis and are equilateral triangles.
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7.2A: Volume using Cross Sections

24. 7.2A: Volume using Cross Sections
Your Turn
Find the volume of the solid whose base is the graphs, 𝒚=𝒙 and 𝒚 𝟐 =𝒙 and the cross sections taken are perpendicular to the 𝒙−axis are semicircles.
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7.2A: Volume using Cross Sections

25. 7.2A: Volume using Cross Sections
Steps
Find the intersection points
Determine the cross section and what axis is it perpendicular to
Apply the PROPER cross section
Evaluate
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7.2A: Volume using Cross Sections

26. 7.2A: Volume using Cross Sections
Example 4
Find the volume of the solid whose base is the region inside the circle, 𝒙 𝟐 + 𝒚 𝟐 =𝟒 and the cross sections taken are perpendicular to the 𝒙−axis are squares.
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7.2A: Volume using Cross Sections

27. 7.2A: Volume using Cross Sections
Example 4
Find the volume of the solid whose base is the region inside the circle, 𝒙 𝟐 + 𝒚 𝟐 =𝟒 and the cross sections taken are perpendicular to the 𝒙−axis are squares.
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7.2A: Volume using Cross Sections

28. 7.2A: Volume using Cross Sections
Example 5
The base of a solid is the region enclosed by a circle centered at the origin and the radius is 5 inches. Setup the equation to find the volume of the solid if all of the cross sections are perpendicular to the 𝒙-axis are squares. Then, use a calculator to solve for the volume.
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7.2A: Volume using Cross Sections

29. 7.2A: Volume using Cross Sections
Example 5 (calc)
The base of a solid is the region enclosed by a circle centered at the origin and the radius is 5 inches. Setup the equation to find the volume of the solid if all of the cross sections are perpendicular to the 𝒙-axis are squares. Then, use a calculator to solve for the volume.
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7.2A: Volume using Cross Sections

30. 7.2A: Volume using Cross Sections
Your Turn
A solid has as its base the region in the 𝒙𝒚-plane bounded by the graph of 𝒙 𝟐 + 𝒚 𝟐 =𝟗. Find the volume of the solid if every cross section by a plane perpendicular to the 𝒙-axis is an equilateral triangle with base in the 𝒙𝒚-plane.
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7.2A: Volume using Cross Sections

31. 7.2A: Volume using Cross Sections
Example 6 (Non-Calc)
Let 𝑹 be the region in the first quadrant bounded by the graph of 𝒚=𝟐 𝒙 , the horizontal line 𝒚=𝟔 and the 𝒚−axis. For each 𝒚, where 𝟎≤𝒚≤𝟔, the e cross section of the solid taken perpendicular to the 𝒚-axis is a rectangle whose height is 3 times the length of its base in region 𝑹. Write, but do not evaluate, an integral expression that gives the volume of the solid.
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7.2A: Volume using Cross Sections

32. 7.2A: Volume using Cross Sections
Example 6 (Non-Calc)
Let 𝑹 be the region in the first quadrant bounded by the graph of 𝒚=𝟐 𝒙 , the horizontal line 𝒚=𝟔 and the 𝒚−axis. For each 𝒚, where 𝟎≤𝒚≤𝟔, the cross section of the solid taken perpendicular to the 𝒚-axis is a rectangle whose height is 3 times the length of its base in region 𝑹. Write, but do not evaluate, an integral expression that gives the volume of the solid.
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7.2A: Volume using Cross Sections

33. 7.2A: Volume using Cross Sections
Example 7 (calc)
Find the volume of the solid whose base is the region is bounded by, 𝒙 𝟐 =𝟏𝟔𝒚 and 𝒚=𝟐. Cross section by a plane perpendicular to the 𝒙-axis are rectangles whose height are TWICE the side of the base in the 𝒙𝒚-plane.
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7.2A: Volume using Cross Sections

34. 7.2A: Volume using Cross Sections
Example 7 (calc)
Find the volume of the solid whose base is the region is bounded by, 𝒙 𝟐 =𝟏𝟔𝒚 and 𝒚=𝟐. Cross section by a plane perpendicular to the 𝒙-axis are rectangles whose height are TWICE the side of the base in the 𝒙𝒚-plane.
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7.2A: Volume using Cross Sections

35. 7.2A: Volume using Cross Sections
Example 7 (calc)
Find the volume of the solid whose base is the region is bounded by, 𝒙 𝟐 =𝟏𝟔𝒚 and 𝒚=𝟐. Cross section by a plane perpendicular to the 𝒙-axis are rectangles whose height are TWICE the side of the base in the 𝒙𝒚-plane.
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7.2A: Volume using Cross Sections

36. 7.2A: Volume using Cross Sections
Example 8 (calc)
Let 𝑹 be the region bounded by the graph, 𝒚= 𝐥𝐧 𝒙 𝟐 +𝟏 , the horizontal line of 𝒚=𝟑 and vertical line of 𝒙=𝟏. Cross sections are perpendicular to the 𝒙-axis is a triangle that is twice the length of the base. Find the volume.
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7.2A: Volume using Cross Sections

37. 7.2A: Volume using Cross Sections
Example 8 (calc)
Let 𝑹 be the region bounded by the graph, 𝒚= 𝐥𝐧 𝒙 𝟐 +𝟏 , the horizontal line of 𝒚=𝟑 and vertical line of 𝒙=𝟏. Cross sections are perpendicular to the 𝒙-axis is a triangle that is twice the length of the base. Find the volume.
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7.2A: Volume using Cross Sections

38. 7.2A: Volume using Cross Sections
Your Turn
Find the volume of the solid that lies between planes of 𝒚= 𝒙 , 𝒚= 𝒆 −𝒙 and the 𝒚−𝒂𝒙𝒊𝒔. The cross sections perpendicular to the 𝒙-axis is a semicircle.
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7.2A: Volume using Cross Sections

39. 7.2A: Volume using Cross Sections
Assignment
Worksheet
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7.2A: Volume using Cross Sections