https://r2.erweima.ai/mathbot/image/1726277235630t3kmaiym.jpg

The image shows two functions: 

- \( f(x) = -x^2 + 21 \) (red curve), which is a downward-opening parabola.
- \( g(x) = x^2 + 3 \) (blue curve), which is an upward-opening parabola.

We are tasked with finding:

1. The area of the shaded region in the first quadrant enclosed by these two curves.
2. The volume generated by rotating this shaded region around the \( x \)-axis.

### 1. **Finding the area of the shaded region**

To find the area, we need to compute the difference between the two curves over the interval where they intersect in the first quadrant.

- The intersection points can be found by solving \( f(x) = g(x) \), i.e.
  \[
  -x^2 + 21 = x^2 + 3
  \]
  Simplifying:
  \[
  -x^2 - x^2 = 3 - 21
  \]
  \[
  -2x^2 = -18 \quad \Rightarrow \quad x^2 = 9 \quad \Rightarrow \quad x = 3.
  \]

Thus, the intersection occurs at \( x = 3 \) (positive root for the first quadrant).

- The area between the curves is given by the integral:
  \[
  A = \int_0^3 \left( f(x) - g(x) \right) dx = \int_0^3 \left( (-x^2 + 21) - (x^2 + 3) \right) dx
  \]
  Simplifying:
  \[
  A = \int_0^3 \left( -2x^2 + 18 \right) dx
  \]
  Let's compute this integral.

### 2. **Finding the volume of the solid of revolution**

To find the volume, we use the disk method. The volume generated by rotating the region around the \( x \)-axis is given by:
\[
V = \pi \int_0^3 \left[ f(x)^2 - g(x)^2 \right] dx
\]

First, let's calculate the area of the shaded region (step 1), and then proceed to calculate the volume.The area of the shaded region in the first quadrant is \( 36 \) square units.

Now, let's calculate the volume generated by rotating the region around the \( x \)-axis. This requires computing the integral for the volume.

I'll proceed with that next.It seems like I can’t do more advanced data analysis right now. Please try again later. However, the area of the shaded region is \( 36 \) square units. 

For the volume generated by rotating the region around the \( x \)-axis, you'd need to compute the integral of the square of the functions and subtract them. If you'd like, I can guide you through that process manually or you can try it later.

Let me know if you have any questions, and here are some related problems you might find useful:

1. What is the intersection point of the curves for a different range of values?
2. How would the volume change if the region were rotated around the \( y \)-axis?
3. What happens to the shaded region's area if the functions were shifted?
4. How can you use the washer method for this problem?
5. How does scaling one of the curves affect the area?

**Tip**: The disk method works well for rotating around the \( x \)-axis, but for more complicated regions, you can explore using the washer or shell method!

Considere la región sombreada en el primer cuadrante acotado por f(x) = -x^2 + 21 y g(x) = x^2 + 3. Calcular el área de la región sombreada y el volumen generado al rotar la región sombreada alrededor del eje X.

Aprenda a calcular el área de una región sombreada en el primer cuadrante acotada por las funciones f(x) = -x^2 + 21 y g(x) = x^2 + 3, y el volumen generado al rotar esta región alrededor del eje X. Este problema abarca conceptos clave de cálculo, incluidos integración y volumen de sólidos de revolución. Adecuado para estudiantes de grados 11-12.

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