Use a triple integral to find the volume of the wedge bounded by the parabolic cylinder yequalsxsquared and the planes zequals18minusy and zequals0.
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Part 1
The volume of the wedge is

enter your response here
cubic units.
square units.
units.
cubic units.
​(Type an exact​ answer.)

We are tasked with finding the volume of a wedge bounded by the parabolic cylinder \( y = x^2 \), the plane \( z = 18 - y \), and the plane \( z = 0 \). To do this, we'll set up a triple integral in the order \( dz\,dy\,dx \).

### Step 1: Determine the limits of integration

1. **For \( z \):**  
   The wedge is bounded by the planes \( z = 0 \) and \( z = 18 - y \), so the limits for \( z \) are:
   \[
   0 \leq z \leq 18 - y
   \]

2. **For \( y \):**  
   Since \( y = x^2 \), we need to determine the range for \( y \). The boundary of \( y \) is from 0 to 18 because the plane \( z = 18 - y \) implies that \( y \leq 18 \). Hence, the limits for \( y \) are:
   \[
   0 \leq y \leq 18
   \]

3. **For \( x \):**  
   Since \( y = x^2 \), for each \( y \), \( x \) ranges from \( -\sqrt{y} \) to \( \sqrt{y} \). So the limits for \( x \) are:
   \[
   -\sqrt{y} \leq x \leq \sqrt{y}
   \]

### Step 2: Set up the triple integral

The volume \( V \) is given by the triple integral:

\[
V = \int_{y=0}^{18} \int_{x=-\sqrt{y}}^{\sqrt{y}} \int_{z=0}^{18-y} dz \, dx \, dy
\]

### Step 3: Evaluate the integrals

1. **Integrating with respect to \( z \):**
   \[
   \int_{0}^{18-y} dz = (18 - y) - 0 = 18 - y
   \]

   So the integral becomes:
   \[
   V = \int_{y=0}^{18} \int_{x=-\sqrt{y}}^{\sqrt{y}} (18 - y) \, dx \, dy
   \]

2. **Integrating with respect to \( x \):**
   Since the integrand \( 18 - y \) does not depend on \( x \), we can integrate \( x \) directly:
   \[
   \int_{-\sqrt{y}}^{\sqrt{y}} dx = 2\sqrt{y}
   \]

   Now the integral becomes:
   \[
   V = \int_{y=0}^{18} (18 - y) \cdot 2\sqrt{y} \, dy
   \]
   \[
   V = 2 \int_{y=0}^{18} (18 - y) \sqrt{y} \, dy
   \]

3. **Integrating with respect to \( y \):**

We now need to integrate \( (18 - y) \sqrt{y} \). We can break this into two terms:
\[
V = 2 \left( 18 \int_{y=0}^{18} \sqrt{y} \, dy - \int_{y=0}^{18} y^{3/2} \, dy \right)
\]

- For the first term:
  \[
  \int \sqrt{y} \, dy = \frac{2}{3} y^{3/2}
  \]
  So:
  \[
  18 \int_{0}^{18} \sqrt{y} \, dy = 18 \cdot \frac{2}{3} \cdot 18^{3/2} = 12 \cdot 18^{3/2}
  \]

- For the second term:
  \[
  \int y^{3/2} \, dy = \frac{2}{5} y^{5/2}
  \]
  So:
  \[
  \int_{0}^{18} y^{3/2} \, dy = \frac{2}{5} \cdot 18^{5/2}
  \]

Now substituting back, we get:
\[
V = 2 \left( 12 \cdot 18^{3/2} - \frac{2}{5} \cdot 18^{5/2} \right)
\]

### Step 4: Simplify and find the volume

- \( 18^{3/2} = 18 \cdot \sqrt{18} = 18 \cdot 4.2426 \approx 76.3668 \)
- \( 18^{5/2} = 18^2 \cdot \sqrt{18} = 324 \cdot 4.2426 \approx 1374.6 \)

So:
\[
V = 2 \left( 12 \cdot 76.3668 - \frac{2}{5} \cdot 1374.6 \right)
\]
\[
V \approx 2 \left( 916.4016 - 549.84 \right)
\]
\[
V \approx 2 \times 366.5616 = 733.1232
\]

Thus, the volume of the wedge is approximately \( \boxed{733.1232} \) cubic units.

Would you like more details, or have any questions?

### Related Questions:
1. How would the volume change if the plane equation was \( z = 20 - y \)?
2. Can you set up the same integral using cylindrical coordinates?
3. What would be the volume if the cylinder was defined as \( y = 2x^2 \)?
4. How does the symmetry in \( x \) affect the limits and computation of this volume?
5. How do you generalize the setup of a triple integral for finding volumes in different regions?

### Tip:
When setting up triple integrals, carefully analyze the boundaries of each variable to ensure the integration limits are correct.

Use a triple integral to find the volume of the wedge bounded by the parabolic cylinder y = x^2 and the planes z = 18 - y and z = 0.

Discover how to find the volume of a wedge formed by the parabolic cylinder y = x^2 and the planes z = 18 - y and z = 0 using triple integrals. This step-by-step guide explains key calculus concepts, such as setting up and evaluating triple integrals, suitable for undergraduate students studying Calculus III.

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