Answer :
We want to find the volume of the solid obtained by rotating the region bounded by
[tex]$$
y = x \quad \text{and} \quad y = 2x^2
$$[/tex]
about the line
[tex]$$
y = 2.
$$[/tex]
Step 1. Find the Points of Intersection
Set
[tex]$$
x = 2x^2.
$$[/tex]
Rearrange and factor:
[tex]$$
2x^2 - x = 0 \quad \Longrightarrow \quad x(2x - 1) = 0.
$$[/tex]
Thus, the intersection points occur when
[tex]$$
x = 0 \quad \text{or} \quad 2x - 1 = 0 \; (x = 0.5).
$$[/tex]
Step 2. Set Up the Washer Method
When the region is rotated about [tex]$y=2$[/tex], we use washers. For a fixed [tex]$x$[/tex] between [tex]$0$[/tex] and [tex]$0.5$[/tex], the distances (radii) from the line [tex]$y=2$[/tex] to the curves are:
- Outer radius, [tex]$R(x)$[/tex]: This is the distance from the line [tex]$y=2$[/tex] to the lower curve [tex]$y=2x^2$[/tex], so
[tex]$$
R(x) = 2 - 2x^2.
$$[/tex]
- Inner radius, [tex]$r(x)$[/tex]: This is the distance from the line [tex]$y=2$[/tex] to the upper curve [tex]$y=x$[/tex], so
[tex]$$
r(x) = 2 - x.
$$[/tex]
The area of a washer is then:
[tex]$$
A(x) = \pi \Bigl( [R(x)]^2 - [r(x)]^2 \Bigr).
$$[/tex]
Step 3. Express the Radii Squared and Their Difference
Compute each squared term:
[tex]$$
[R(x)]^2 = (2 - 2x^2)^2 = 4 - 8x^2 + 4x^4,
$$[/tex]
[tex]$$
[r(x)]^2 = (2 - x)^2 = 4 - 4x + x^2.
$$[/tex]
Subtract to find the difference:
[tex]$$
[R(x)]^2 - [r(x)]^2 = \bigl(4 - 8x^2 + 4x^4\bigr) - \bigl(4 - 4x + x^2\bigr) = 4x - 9x^2 + 4x^4.
$$[/tex]
Step 4. Set Up and Evaluate the Integral
The volume [tex]$V$[/tex] is given by:
[tex]$$
V = \pi \int_{0}^{0.5} \Bigl(4x - 9x^2 + 4x^4\Bigr)\,dx.
$$[/tex]
Integrate term-by-term:
- For [tex]$4x$[/tex]:
[tex]$$
\int 4x\,dx = 2x^2.
$$[/tex]
- For [tex]$-9x^2$[/tex]:
[tex]$$
\int -9x^2\,dx = -3x^3.
$$[/tex]
- For [tex]$4x^4$[/tex]:
[tex]$$
\int 4x^4\,dx = \frac{4}{5}x^5.
$$[/tex]
Thus, the antiderivative is:
[tex]$$
F(x) = 2x^2 - 3x^3 + \frac{4}{5}x^5.
$$[/tex]
Now, evaluate [tex]$F(x)$[/tex] from [tex]$x=0$[/tex] to [tex]$x=0.5$[/tex]:
At [tex]$x = 0.5$[/tex]:
[tex]$$
F(0.5) = 2(0.5)^2 - 3(0.5)^3 + \frac{4}{5}(0.5)^5.
$$[/tex]
Calculate each term:
- [tex]$2(0.5)^2 = 2(0.25) = 0.5$[/tex],
- [tex]$-3(0.5)^3 = -3(0.125) = -0.375$[/tex],
- [tex]$\frac{4}{5}(0.5)^5 = \frac{4}{5} \times \frac{1}{32} = 0.025.$[/tex]
Thus,
[tex]$$
F(0.5) = 0.5 - 0.375 + 0.025 = 0.15.
$$[/tex]
The integral becomes:
[tex]$$
\int_{0}^{0.5} \Bigl(4x - 9x^2 + 4x^4\Bigr)\,dx = 0.15.
$$[/tex]
Step 5. Multiply by [tex]$\pi$[/tex] to Determine the Volume
The final volume is:
[tex]$$
V = \pi \times 0.15 = \frac{3\pi}{20}.
$$[/tex]
Answer: The volume of the solid is [tex]$\boxed{\frac{3}{20} \pi}$[/tex] cubic units.
[tex]$$
y = x \quad \text{and} \quad y = 2x^2
$$[/tex]
about the line
[tex]$$
y = 2.
$$[/tex]
Step 1. Find the Points of Intersection
Set
[tex]$$
x = 2x^2.
$$[/tex]
Rearrange and factor:
[tex]$$
2x^2 - x = 0 \quad \Longrightarrow \quad x(2x - 1) = 0.
$$[/tex]
Thus, the intersection points occur when
[tex]$$
x = 0 \quad \text{or} \quad 2x - 1 = 0 \; (x = 0.5).
$$[/tex]
Step 2. Set Up the Washer Method
When the region is rotated about [tex]$y=2$[/tex], we use washers. For a fixed [tex]$x$[/tex] between [tex]$0$[/tex] and [tex]$0.5$[/tex], the distances (radii) from the line [tex]$y=2$[/tex] to the curves are:
- Outer radius, [tex]$R(x)$[/tex]: This is the distance from the line [tex]$y=2$[/tex] to the lower curve [tex]$y=2x^2$[/tex], so
[tex]$$
R(x) = 2 - 2x^2.
$$[/tex]
- Inner radius, [tex]$r(x)$[/tex]: This is the distance from the line [tex]$y=2$[/tex] to the upper curve [tex]$y=x$[/tex], so
[tex]$$
r(x) = 2 - x.
$$[/tex]
The area of a washer is then:
[tex]$$
A(x) = \pi \Bigl( [R(x)]^2 - [r(x)]^2 \Bigr).
$$[/tex]
Step 3. Express the Radii Squared and Their Difference
Compute each squared term:
[tex]$$
[R(x)]^2 = (2 - 2x^2)^2 = 4 - 8x^2 + 4x^4,
$$[/tex]
[tex]$$
[r(x)]^2 = (2 - x)^2 = 4 - 4x + x^2.
$$[/tex]
Subtract to find the difference:
[tex]$$
[R(x)]^2 - [r(x)]^2 = \bigl(4 - 8x^2 + 4x^4\bigr) - \bigl(4 - 4x + x^2\bigr) = 4x - 9x^2 + 4x^4.
$$[/tex]
Step 4. Set Up and Evaluate the Integral
The volume [tex]$V$[/tex] is given by:
[tex]$$
V = \pi \int_{0}^{0.5} \Bigl(4x - 9x^2 + 4x^4\Bigr)\,dx.
$$[/tex]
Integrate term-by-term:
- For [tex]$4x$[/tex]:
[tex]$$
\int 4x\,dx = 2x^2.
$$[/tex]
- For [tex]$-9x^2$[/tex]:
[tex]$$
\int -9x^2\,dx = -3x^3.
$$[/tex]
- For [tex]$4x^4$[/tex]:
[tex]$$
\int 4x^4\,dx = \frac{4}{5}x^5.
$$[/tex]
Thus, the antiderivative is:
[tex]$$
F(x) = 2x^2 - 3x^3 + \frac{4}{5}x^5.
$$[/tex]
Now, evaluate [tex]$F(x)$[/tex] from [tex]$x=0$[/tex] to [tex]$x=0.5$[/tex]:
At [tex]$x = 0.5$[/tex]:
[tex]$$
F(0.5) = 2(0.5)^2 - 3(0.5)^3 + \frac{4}{5}(0.5)^5.
$$[/tex]
Calculate each term:
- [tex]$2(0.5)^2 = 2(0.25) = 0.5$[/tex],
- [tex]$-3(0.5)^3 = -3(0.125) = -0.375$[/tex],
- [tex]$\frac{4}{5}(0.5)^5 = \frac{4}{5} \times \frac{1}{32} = 0.025.$[/tex]
Thus,
[tex]$$
F(0.5) = 0.5 - 0.375 + 0.025 = 0.15.
$$[/tex]
The integral becomes:
[tex]$$
\int_{0}^{0.5} \Bigl(4x - 9x^2 + 4x^4\Bigr)\,dx = 0.15.
$$[/tex]
Step 5. Multiply by [tex]$\pi$[/tex] to Determine the Volume
The final volume is:
[tex]$$
V = \pi \times 0.15 = \frac{3\pi}{20}.
$$[/tex]
Answer: The volume of the solid is [tex]$\boxed{\frac{3}{20} \pi}$[/tex] cubic units.
