To find the volume of the solid bounded by the parabolic cylinder $z = 4 - x^2$ and the plane $y = 5$ in the first octant, we need to set up a triple integral over the region defined by these surfaces.

Step 1: Understand the boundaries

• The equation $z = 4 - x^2$ describes a parabolic cylinder.
• The plane $y = 5$ means $y$ ranges from 0 to 5.
• The first octant restricts the coordinates to $x \geq 0$, $y \geq 0$, and $z \geq 0$.

Since $z = 4 - x^2$, the value of $x$ must satisfy $4 - x^2 \geq 0$, which implies $x$ ranges from 0 to 2.

Step 2: Set up the triple integral

The volume is given by the triple integral:

$V = \int_0^5 \int_0^2 \int_0^{4 - x^2} dz \, dx \, dy$

Where:

• $y$ ranges from 0 to 5.
• $x$ ranges from 0 to 2 (based on the parabolic cylinder).
• $z$ ranges from 0 to $4 - x^2$ for each point in the $x, y$-plane.

Step 3: Perform the integration

1. Integrate with respect to $z$:

$\int_0^{4 - x^2} dz = (4 - x^2)$

2. Integrate with respect to $x$:

$\int_0^2 (4 - x^2) dx$

This is a straightforward integral:

$\int_0^2 (4 - x^2) dx = \left[ 4x - \frac{x^3}{3} \right]_0^2$

Evaluating:

= 8 - \frac{8}{3} = \frac{24}{3} - \frac{8}{3} = \frac{16}{3}$$ 3. **Integrate with respect to $$y$$:** $$\int_0^5 \frac{16}{3} dy = \frac{16}{3} \cdot 5 = \frac{80}{3}$$ ### Final answer: The volume of the solid is $$\frac{80}{3}$$ cubic units. --- Would you like a more detailed explanation of any of the steps or have further questions? Here are some related questions to consider: 1. How would the volume change if the parabolic cylinder were defined by $$z = 9 - x^2$$? 2. What if the plane were given by $$y = 3$$ instead of $$y = 5$$? 3. How would the volume calculation differ if we worked in the second octant instead of the first octant? 4. How does the concept of parabolic cylinders apply to other types of solid volume calculations? 5. Could we use a different coordinate system, such as cylindrical coordinates, to solve this problem? **Tip:** When dealing with volume problems, always check the bounds carefully—ensuring all limits are properly derived from the equations of the surfaces involved!