To find the value of $c$ for the integral that calculates the volume of the solid bounded by the surface $f(x, y) = 2x^2 + 4y^2 + 1$ above and the $xy$-plane below, we can set up the double integral over the given region.

Step 1: Understand the integral expression

The volume $V$ of the solid is found using the double integral: $V = \iint_R f(x, y) \, dx \, dy$ where $R$ is the region defined by the bounds $4 \leq x \leq 7$ and $-6 \leq y \leq 16$.

Step 2: Express the integral

The integral to calculate the volume is: $V = \int_{y=-6}^{16} \int_{x=4}^{7} (2x^2 + 4y^2 + 1) \, dx \, dy$

Step 3: Identify the variable limits

The integral is set up as follows: $\int_{y=-6}^{16} \left( \int_{x=4}^{7} (2x^2 + 4y^2 + 1) \, dx \right) dy$ Here, the limits for $x$ are from $4$ to $7$, and the limits for $y$ are from $-6$ to $16$.

Step 4: Identify the constant c in the context of this problem

The problem asks for the value of $c$ in the integral. From the context and the choices given:

• The integral represents a region on the $xy$-plane.
• Typically, the value $c$ in an integral like this refers to the limits of integration.

Given the integral setup:

• $c$ for the lower bound of $y$ is $-6$.
• $c$ for the upper bound of $y$ is $16$.
• $c$ for the lower bound of $x$ is $4$.
• $c$ for the upper bound of $x$ is $7$.

Since you provided the following choices (4, -6, 7, 16), all these values are correct and correspond to the integration limits. However, if you need to select a specific value from these choices, you would need to specify which bound is being asked for.

Final Answer:

The correct value of $c$ depends on which specific limit of integration you are asked for. Based on the problem as presented:

• $c = 4$ for the lower limit of $x$,
• $c = -6$ for the lower limit of $y$,
• $c = 7$ for the upper limit of $x$,
• $c = 16$ for the upper limit of $y$.

Would you like to specify which $c$ you are asking for, or need further clarification? Here are 5 related questions:

1. How do you calculate the volume of a solid bounded by a function and a plane?
2. What is the significance of the order of integration in double integrals?
3. How would the problem change if the region of integration were circular?
4. What is the role of $f(x, y)$ in the integral for volume?
5. How do you solve double integrals using polar coordinates?

Tip: When dealing with multiple integrals, always make sure the limits of integration match the region described by the problem.