Volumes of revolution

Given that, for n

show that

Figure 1 Figure 2

Figure 1 shows the central vertical cross-section of a solid wooden ornament.

Figure 2 shows the curve with equation

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation y = and the y-axis.

The ornament is modelled by the solid of revolution formed when R is rotated 360° about the y-axis. The units are centimetres.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of wood needed to make the ornament, according to the model. Give your answer to 2 significant figures.

[ Solutions based entirely on calculator technology are not acceptable.]

Given that

• the density of the wood is 0.85 g/cm³

• the mass of the ornament is 6 grams

comment on the suitability of the model.

Given

show that

x = –4 x = 4

Figure 1 Figure 2

Figure 1 shows a solid paperweight with a flat base.

Figure 2 shows the curve with equation

where H is a positive constant and x is in radians.

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = –4, the line with equation x = 4 and the x-axis.

The paperweight is modelled by the solid of revolution formed when R is rotated 180° about the x-axis.

Given that the maximum height of the paperweight is 2 cm,

write down the value of H.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of the paperweight, according to the model. Give your answer to 2 decimal places.

[ Solutions based entirely on calculator technology are not acceptable.]

State a limitation of the model.

Given

show that

x = –4 x = 4

Figure 1
Figure 2

Figure 1 shows a solid paperweight with a flat base.

Figure 2 shows the curve with equation

where H is a positive constant and x is in radians.

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = –4, the line with equation x = 4 and the x-axis.

The paperweight is modelled by the solid of revolution formed when R is rotated 180° about the x-axis.

Given that the maximum height of the paperweight is 2 cm,

write down the value of H.

Using algebraic integration and the result in part (a), determine, in cm³, the volume of the paperweight, according to the model. Give your answer to 2 decimal places.

[ Solutions based entirely on calculator technology are not acceptable.]

State a limitation of the model.

Solutions based entirely on graphical or numerical methods are not acceptable.

Figure 1 shows a sketch of part of the curve with equation

y = arsinhx x 0

and the straight line with equation y = β The line and the curve intersect at the point with coordinates (α, β)

The finite region R, shown shaded in Figure 1, is bounded by the curve with equation y = arsinh x, the y-axis and the line with equation y = β

The region R is rotated through 2π radians about the y-axis.

Use calculus to find the exact value of the volume of the solid generated.

Solutions based entirely on graphical or numerical methods are not acceptable.

Figure 1 shows a sketch of part of the curve with equation

y = arsinhx
x 0

and the straight line with equation y = β The line and the curve intersect at the point with coordinates (α, β)

The finite region R, shown shaded in Figure 1, is bounded by the curve with equation y = arsinh x, the y-axis and the line with equation y = β

The region R is rotated through 2π radians about the y-axis.

Use calculus to find the exact value of the volume of the solid generated.

A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student’s design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the y-axis, the x-axis, the line with equation x = 1, the curve C₁ and the curve C₂ through 360° about the y-axis. The point A has coordinates (1, 0.5) and the point B has coordinates (0.5, 2.5) where the units are centimetres.

The curve C₁ is modelled by the equation

Determine the value of a and the value of b according to the model.

The curve C₂ is modelled to be an arc of the circle with centre (0, 3).

Use calculus to determine the volume of plastic required to make the chess piece according to the model.

A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student’s design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the y-axis, the x-axis, the line with equation x = 1, the curve C₁ and the curve C₂ through 360° about the y-axis. The point A has coordinates (1, 0.5) and the point B has coordinates (0.5, 2.5) where the units are centimetres.

The curve C₁ is modelled by the equation

Determine the value of a and the value of b according to the model.

The curve C₂ is modelled to be an arc of the circle with centre (0, 3).

Use calculus to determine the volume of plastic required to make the chess piece according to the model.