The region, \(R\), is bounded by the function, \(y=x^3\), the \(x\)-axis and the lines \(x=1\) and \(x=2\). What is the volume of the solid of revolution obtained when the region \(R\) is rotated about the \(x\)-axis? (3 marks) --- 6 WORK AREA LINES (style=lined) ---
Show Worked Solution
\(V\)
\(=\pi \displaystyle \int_1^2 y^2\, d x\)
\(=\pi \displaystyle \int_1^2 x^6\,d x\)
\(=\pi\left[\dfrac{x^7}{7}\right]_1^2\)
\(=\pi\left(\dfrac{2^7-1}{7}\right)\)
\(=\dfrac{127 \pi}{7} \ \ \text{u}^3\)
Calculus, EXT1 C3 2021 HSC 13a
A 2-metre-high sculpture is to be made out of concrete. The sculpture is formed by rotating the region between `y = x^2, y = x^2 + 1` and `y = 2` around the `y`-axis.
Find the volume of concrete needed to make the sculpture. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
Show Worked Solution
| `V` | `= pi int_0^2 y\ dy-pi int_1^2 y-1\ dy` |
| `= pi [(y^2)/2]_0^2-pi [(y^2)/2-y]_1^2` | |
| `= pi(2-0)-pi[(2-2)-(1/2-1)]` | |
| `= 2pi-pi(1/2)` | |
| `= (3pi)/2\ text(u³)` |
Calculus, EXT1 C3 SM-Bank 2
The parabola with equation `y = 9 - x^2` cuts the `y`-axis at `P(0,9)` and the `x`-axis at `Q(3,0)`.
Find the exact volume of the solid of revolution formed when the area between the line `PQ` and the parabola is rotated about the `y`-axis. (4 marks)
--- 8 WORK AREA LINES (style=lined) ---
Show Worked Solution
`text(Equation of)\ PQ:`
`m = −3,\ \ ytext(-intercept) = 9`
`y = 9 – 3x`
`text(Rotating about the)\ ytext(-axis:)`
| `x_1^(\ 2)` | `= 9 – y\ \ \ text{(parabola)}` |
| `y` | `= 9 – 3x_2\ \ \ (text{line}\ PQ)` |
| `3x_2` | `= 9 – y` |
| `x_2` | `= 3 – y/3` |
| `x_2^(\ 2)` | `= (3 – y/3)^2` |
| `V` | `= pi int_0^9 x_1^(\ 2) – x_2^(\ 2)\ dy` |
| `= pi int_0^9 9 – y – (3 – y/3)^2\ dy` | |
| `= pi int_0^9 9 – y – (9 – 2y + (y^2)/9)\ dy` | |
| `= pi int_0^9 y – (y^2)/9\ dy` | |
| `= pi [(y^2)/2 – (y^3)/27]_0^9` | |
| `= pi(81/2 – 27)` | |
| `= (27pi)/2\ text(units³)` |
Calculus, EXT1* C3 2019 HSC 13d
The diagram shows the region bounded by the curve `y = x - x^3`, and the `x`-axis between `x = 0` and `x = 1`. The region is rotated about the `x`-axis to form a solid.
Find the exact value of the volume of the solid formed. (3 marks)
Calculus, EXT1* C3 2018 HSC 14b
The shaded region shown in the diagram is bounded by the curve `y = x^4 + 1`, the `y`-axis and the line `y = 10`.
Find the volume of the solid of revolution formed when the shaded region is rotated about the `y`-axis. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
Calculus, EXT1* C3 2007 HSC 9a
The shaded region in the diagram is bounded by the curve `y = x^2 + 1`, the `x`-axis, and the lines `x = 0` and `x = 1.`
Find the volume of the solid of revolution formed when the shaded region is rotated about the `x`-axis. (3 marks)
Calculus, EXT1* C3 2006 HSC 4b
In the diagram, the shaded region is bounded by the parabola `y = x^2 + 1`, the `y`-axis and the line `y = 5`.
Find the volume of the solid formed when the shaded region is rotated about the `y`-axis. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
Calculus, EXT1* C3 2005 HSC 6c
The graphs of the curves `y = x^2` and `y = 12 - 2x^2` are shown in the diagram.
- Find the points of intersection of the two curves. (1 mark)
--- 4 WORK AREA LINES (style=lined) ---
- The shaded region between the curves and the `y`-axis is rotated about the `y`-axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
Calculus, EXT1* C3 2011 HSC 8b
The diagram shows the region enclosed by the parabola `y = x^2`, the `y`-axis and the line `y = h`, where `h > 0`. This region is rotated about the `y`-axis to form a solid called a paraboloid. The point `C` is the intersection of `y = x^2` and `y = h`.
The point `H` has coordinates `(0, h)`.
- Find the exact volume of the paraboloid in terms of `h`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- A cylinder has radius `HC` and height `h`.
What is the ratio of the volume of the paraboloid to the volume of the cylinder? (1 mark)
--- 8 WORK AREA LINES (style=lined) ---
Calculus, EXT1* C3 2013 HSC 15b
The region bounded by the `x`-axis, the `y`-axis and the parabola `y = (x-2)^2` is rotated about the `y`-axis to form a solid.
Find the volume of the solid. (4 marks)
--- 8 WORK AREA LINES (style=lined) ---