Point 3. Volumes

Volume of a body with known areas of its parallel cross-sections

Let some body is situated between the planes , , and for any the area of its cross-section by a plane perpendicular to Ox-axis is known (see fig. 16). The volume of the body equals the Fig. 16 integral . ( 22 )

■ An element of the volume is the volu-me of a right circular cylinder with the base and the altitude , . Adding all the- Fig. 17 se elements we get the required volume represent-ted by the formula (22).■

Ex. 15. Find the volume of the triaxial ellipsoid (fig. 17)

It’s evident that . For any the cross-section of the body perpendicular to the Ox-axis is the ellipse with the semi-axes

and the area

.

Therefore the volume of the ellipsoid by the formula (22) equals

.

Volume of a body of rotation

A curvilinear trapezium (see fig. 1) rotates about -axis. Prove that the volume of the corres-ponding body (body of rotation, fig 18) equals the definite integral

. ( 23 )

Fig. 18 ■For any a cross-section of the body of ro- tation by a plane perpendicular to Ox-axis is a circle of radius (fig. 18). Therefore its area , and by the formula (22) the volume of the body in question is given by the formula (23).■

Let the same curvilinear trapezium (fig. 1) rotates about -axis which doesn’t pass through the interior of the trapezium⁹. Prove that the volume of the body of its rotation is represented by the next integral:

. ( 24 )

Instructions. As an element of the volume one can take the volume of a part of

the body generated by rotation of the rectangle with the sides about the Oy-axis. Then the element of the volume is whence it follows the formula (24).

Ex. 16. Let the arc of a sinusoid rotates about the Ox-, Oy-axes. Calculate the volumes of corresponding bodies of rotation.

With the help of the formulas (23), (24) we get

.

.

A curvilinear trapezium (fig. 6) rotates about -axis. Prove that the volume of the corresponding body of rotation is represented by the integral

. ( 25 )

Ex. 17. An ellipse rotates about the Ox-, Oy-axes. Calculate the volumes of corresponding bodies of rotation.

From the equation of the ellipse

and by the formulas (24), (25) we have

; .

Point 4. Economic applications

Problem 1 (produced quantity). Let the labour productivity of a some factory at a time moment t equals . It is known (see the formula (11) of the point 2 of prce-ding lecture) that its produced quantity U during the time interval [0, T] equals

.

Ex. 18. Let the labour productivity of a factory is . Then its produced quantity

.

Problem 2 (costs of conservation of goods). Let is the quantity of goods in the storage at a time moment t, and a constant quantity h represents the costs of conservation of unit of goods per unit of time. Then the costs of conservation of goods during a time interval (or an element of the costs of conservation) equals

.

Adding all these elements from to we get the costs of conservation of goods during the time interval , that is

.

Let, for example, we study the case of uniform consumption of all goods from at a time to at a time . In this case the quantity of goods at a time mo-ment t is

,

and the costs of conservation of goods equals

.