Calcular el volumen del solido generado al rotar la región limitada por, 14 𝑥2 + 2 Paray = 2y = 9, respecto al eje y, y dibujar su respectiva gráfica.

1. Break Down the Problem

We need to calculate the volume of the solid generated by rotating the region bounded by the equations $14x^2 + 2y = 9$ and $y = 2$ about the y-axis.

2. Relevant Concepts

To find the volume of the solid obtained by revolving a region around the y-axis, we can use the method of cylindrical shells or the disk/washer method. Here, we can express $x$ in terms of $y$ from the equation and then use integration to find the volume.

First, let’s express $x$ from the given equation $14x^2 + 2y = 9$: $14x^2 = 9 - 2y \quad \Rightarrow \quad x^2 = \frac{9 - 2y}{14} \quad \Rightarrow \quad x = \sqrt{\frac{9 - 2y}{14}}$

3. Analysis and Detail

Now, we find the volume using the disk method: $V = \pi \int_{y_1}^{y_2} [R(y)]^2 \, dy$ where $R(y) = x(y)$.

From the equations given:

• $y_1 = 2$
• $y_2$ can be found by setting $14x^2 + 2y = 9$ to $y = 0$: $14x^2 = 9 \quad \Rightarrow \quad x^2 = \frac{9}{14} \quad \Rightarrow \quad y_2 = \frac{9 - 2 \cdot 0}{2} = \frac{9}{2} = 4.5$

Now, we need to integrate from $y = 2$ to $y = 4.5$: $V = \pi \int_{2}^{4.5} \left( \sqrt{\frac{9 - 2y}{14}} \right)^2 \, dy$ $V = \pi \int_{2}^{4.5} \frac{9 - 2y}{14} \, dy$ $V = \frac{\pi}{14} \int_{2}^{4.5} (9 - 2y) \, dy$

Now we calculate the integral: $\int (9 - 2y) \, dy = 9y - y^2$ Evaluating from $2$ to $4.5$: $\left[ 9(4.5) - (4.5)^2 \right] - \left[ 9(2) - (2)^2 \right]$ $= \left[ 40.5 - 20.25 \right] - \left[ 18 - 4 \right]$ $= 20.25 - 14 = 6.25$

Now, substituting back into the volume equation: $V = \frac{\pi}{14} (6.25) = \frac{6.25\pi}{14} = \frac{25\pi}{56}$

4. Verify and Summarize

The volume of the solid generated by rotating the described region about the y-axis is $\frac{25\pi}{56}$.

Final Answer

The volume of the solid generated by rotating the region is $\frac{25\pi}{56}$ cubic units.

Regarding the graphical representation, you can sketch the curves given by the equations $14x^2 + 2y = 9$ and $y = 2$ to visualize the region being rotated.