The parabolic segment

The parabolic segment has a base a = 4 cm and a height v = 6 cm. Calculate the volume of the body that results from the rotation of this segment

a) around its base
b) around its axis.

Correct answer:

V₁ =  51.2 cm³
V₂ =  50.2655 cm³

Step-by-step explanation:

$$\begin{gathered} a=4\text{ cm } \\ v=6\text{ cm } \\ f(x)=q{x}^2 \\ f(a\mathrm{/}2)=q(a\mathrm{/}2{)}^2=v \\ 6=q\cdot 2^2 \\ q=6\mathrm{/}{2}^2={\displaystyle \frac{3}{2}}=1.5 \\ {V}_1={\displaystyle \frac{8}{15}}\cdot v\cdot a^2={\displaystyle \frac{8}{15}}\cdot 6\cdot 4^2={\displaystyle \frac{256}{5}}{\text{cm}}^3=51.2{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} f(x)=v-{\displaystyle \frac{6}{4}}{x}^2 \\ f(x)=6-{\displaystyle \frac{6}{4}}{x}^2 \\ {x}_0=-a\mathrm{/}2=-4\mathrm{/}2=-2 \\ {x}_1=a\mathrm{/}2=4\mathrm{/}2=2 \\ {V}_2=\pi \cdot {\int }_{x_0}^{x_1}f(x)d_x \\ {V}_2=\pi \cdot {\int }_{x_0}^{x_1}({\displaystyle \frac{6}{4}}{x}^2-v)d_x \\ {V}_2=\pi \cdot [{\displaystyle \frac{6}{4}}\cdot x^3\mathrm{/}3-v_x{]}_{x_0}^{x_1} \\ F(x)=6x-{\displaystyle \frac{6}{4}}\cdot x^3\mathrm{/}3 \\ {F}_1=6\cdot x_1-{\displaystyle \frac{6}{4}}\cdot x_1^3\mathrm{/}3=6\cdot 2-{\displaystyle \frac{6}{4}}\cdot 2^3\mathrm{/}3=8 \\ {F}_0=6\cdot x_0-{\displaystyle \frac{6}{4}}\cdot x_0^3\mathrm{/}3=6\cdot (-2)-{\displaystyle \frac{6}{4}}\cdot (-2{)}^3\mathrm{/}3=-8 \\ {V}_2=\pi \cdot (F_1-F_0)=3.1416\cdot (8-(-8))=50.2655{\text{cm}}^3 \end{gathered}$$

Try another example

Related math problems and questions:

• Triangular prism
  The triangular prism has a base in the shape of a right triangle, the legs of which is 9 cm and 40 cm long. The height of the prism is 20 cm. What is its volume cm³? And the surface cm²?
• Cuboid and eq2
  Calculate the volume of cuboid with square base and height 6 cm if the surface area is 48 cm².
• Spherical segment
  Calculate the volume of a spherical segment 18 cm high. The diameter of the lower base is 80 cm, the upper base 60 cm.
• Diameter = height
  The cylinder's surface, the height of which is equal to the diameter of the base, is 4239 cm square. Calculate the cylinder volume.
• Square
  Suppose the square's sides' length decreases by a 25% decrease in the content area of 28 cm². Determine the side length of the original square.
• Quadrangular pyramid
  Given is a regular quadrangular pyramid with a square base. The body height is 30 cm and volume V = 1000 cm³. Calculate its side a and its surface area.
• Triangular pyramid
  A regular tetrahedron is a triangular pyramid whose base and walls are identical equilateral triangles. Calculate the height of this body if the edge length is a = 8 cm
• Truncated pyramid
  The truncated regular quadrilateral pyramid has a volume of 74 cm³, a height v = 6 cm, and an area of the lower base 15 cm² greater than the upper base's content. Calculate the area of the upper base.
• Rotation
  The right triangle with legs 11 cm and 18 cm rotate around the longer leg. Calculate the volume and surface area of the formed cone.
• Uboid volume
  Calculate the cuboid volume if the walls are 30cm², 35cm², 42cm²
• Equilateral cylinder
  Equilateral cylinder (height = base diameter; h = 2r) has a volume of V = 199 cm³ . Calculate the surface area of the cylinder.
• Regular square prism
  The volume of a regular square prism is 192 cm³. The size of its base edge and the body height is 1: 3. Calculate the surface of the prism.
• Trapezoid
  Area of trapezoid is 135 cm². Sides a, c and height h are in a ratio 6:4:3. How long are a,c and h? Make calculation...
• Body diagonal
  The cuboid has a volume of 32 cm³. Its side surface area is double as one of the square bases. What is the length of the body diagonal?
• Calculate
  Calculate the cone's surface and volume that results from the rotation of the right triangle ABC with the squares 6 cm and 9 cm long around the shorter squeegee.
• Rhombus base
  Calculate the volume and surface area of prisms whose base is a rhombus with diagonals u₁ = 12 cm and u₂ = 15 cm. Prism height is twice the base edge length.
• Secret treasure
  Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Find the container's radius r (and height h) so that they can hide the largest possible treasure.