Answer
The volume of the solid is \( \frac{59049\pi}{4} \).
Solution
To set up the definite integral using the shell method for the given curves \( y = x^{3/2} \) and \( y = 27 \), we need to follow these steps:
1. **Identify the region of integration**: We are looking for the area between the curve \( y = x^{3/2} \) and the line \( y = 27 \) from \( x = 0 \) to the point where they intersect.
2. **Find the intersection points**: Set \( x^{3/2} = 27 \) to find the value of \( x \) at which the two curves intersect.
3. **Set up the shell method integral**: The shell method involves integrating with respect to \( y \). The formula for the shell method is:
\[
V = 2\pi \int_{a}^{b} (radius)(height) \, dy
\]
where the radius is the distance from the axis of rotation (in this case, the y-axis) and the height is the difference between the outer and inner functions.
4. **Evaluate the integral**.
Let's go through these steps in detail.
### Step 1: Find the intersection points
Set \( x^{3/2} = 27 \):
\[
x^{3/2} = 27
\]
To solve for \( x \), raise both sides to the power of \( \frac{2}{3} \):
\[
x = 27^{2/3}
\]
Calculating \( 27^{2/3} \):
\[
27^{2/3} = (3^3)^{2/3} = 3^2 = 9
\]
So, the curves intersect at \( x = 9 \).
### Step 2: Set up the shell method integral
The radius of the shell is \( y \) (the distance from the y-axis), and the height of the shell is given by the difference between the line and the curve:
\[
\text{Height} = 27 - y^{2/3}
\]
Thus, the integral becomes:
\[
V = 2\pi \int_{0}^{27} y \cdot (27 - y^{2/3}) \, dy
\]
### Step 3: Evaluate the integral
Now we will evaluate the integral:
\[
V = 2\pi \int_{0}^{27} (27y - y^{5/3}) \, dy
\]
Let's calculate this integral.
1. Calculate the integral of \( 27y \):
\[
\int 27y \, dy = \frac{27y^2}{2}
\]
2. Calculate the integral of \( y^{5/3} \):
\[
\int y^{5/3} \, dy = \frac{y^{8/3}}{8/3} = \frac{3}{8}y^{8/3}
\]
Now, we can combine these results and evaluate from \( 0 \) to \( 27 \):
\[
V = 2\pi \left[ \left( \frac{27y^2}{2} - \frac{3}{8}y^{8/3} \right) \bigg|_{0}^{27} \right]
\]
Calculating at the upper limit \( y = 27 \):
\[
\frac{27(27^2)}{2} - \frac{3}{8}(27^{8/3})
\]
Calculating \( 27^2 = 729 \) and \( 27^{8/3} = (27^{2/3})^4 = 9^4 = 6561 \):
\[
= \frac{27 \cdot 729}{2} - \frac{3}{8} \cdot 6561
\]
Calculating \( \frac{27 \cdot 729}{2} = \frac{19683}{2} \) and \( \frac{3 \cdot 6561}{8} = \frac{19683}{8} \):
\[
= \frac{19683}{2} - \frac{19683}{8}
\]
Finding a common denominator (8):
\[
= \frac{19683 \cdot 4}{8} - \frac{19683}{8} = \frac{19683 \cdot 3}{8} = \frac{59049}{8}
\]
Thus, the volume \( V \) is:
\[
V = 2\pi \cdot \frac{59049}{8} = \frac{118098\pi}{8} = \frac{59049\pi}{4}
\]
### Final Answer
The volume of the solid formed by rotating the region around the y-axis is:
\[
V = \frac{59049\pi}{4}
\]
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