10.5.2 An Example of Finding Cross-Sectional Volumes

Question 1

An Example of Finding Cross-Sectional Volumes

Answer 1

• The volume of a solid with vertical cross-sections of area A(x) is V, where
.
• The volume of a solid with horizontal cross-sections of area A(y) is V, where
.

Question 2

note

Answer 2

• Sometimes you might be given the description of a region in space, and then you will be asked to find the volume. It is not a bad idea to draw the region in question. By drawing the region, you’ll discover such things as how the cross-sections are defined, as well as what interval the region lies within.
• Notice that this drawing is done from a perspective
  viewpoint. Perspective views are good for illustrating how an object appears three-dimensionally, since viewing the object from above might not reveal enough about its shape.
• Since the cross-sections are defined as perfect squares, the first step in finding the volume is to determine the length of the side of the square. The base of each square is equal to the difference of the two functions.
• Once you know the length, finding an equation for the area is simple. The cross-sections are defined to be squares. So the area of a cross-section is the square of the length of its base.
• Integrating the square of the base along the given interval gives you the volume.
• Watch out! When evaluating definite integrals, it is very easy to make algebraic mistakes. Watch out for them.

Question 3

What is the volume of this solid? The base of the solid is bounded by the curves f (x) = 1 − 2x and g (x) = x 2 − 2, and the cross-sections perpendicular to the x‑axis are semicircles.

Answer 3

64π/15

Question 4

What is the volume of this solid? The base of the solid is bounded by the x‑axis, the y‑axis, and the line y = 3 − x, and the cross-sections are isosceles right triangles perpendicular to the x‑axis.

Answer 4

9/2

Question 5

The volume of a solid is independent of _______.

Answer 5

the direction of the slices

Question 6

What is the volume of this solid? The base of the solid is bounded by the curves f (x) = 1 − 2x and g (x) = x 2 − 2, and the cross-sections perpendicular to the x‑axis are perfect squares.

Answer 6

512/15

Question 7

What is the volume of this solid? The base of the solid is bounded by the curves f (x) = x 2 and g (x) = x + 2, and the cross-sections perpendicular to the x‑axis are rectangles of height 1.

Answer 7

9/2

Question 8

What is the volume of this solid? The base of the solid is bounded by the curves f (x) = x 2 and g (x) = x + 2, and the cross-sections are equilateral triangles.

Answer 8

81√3/40