0≤r≤2 and π/2≤¢≤3π/4 shade the polar region of given polar equation ordered anticlockwise 

To shade the polar region defined by the inequalities \( 0 \leq r \leq 2 \) and \( \frac{\pi}{2} \leq \theta \leq \frac{3\pi/4} \), follow these steps:

1. **Understand the region**: 
   - \( 0 \leq r \leq 2 \) means that the radial distance \( r \) from the origin (pole) can range from 0 to 2.
   - \( \frac{\pi}{2} \leq \theta \leq \frac{3\pi/4} \) indicates that the angle \( \theta \) lies between \( \frac{\pi}{2} \) (90 degrees) and \( \frac{3\pi/4} \) (135 degrees).

2. **Draw the polar grid**:
   - Start by drawing the polar axes with the origin at the center.
   - Mark the angles \( \frac{\pi}{2} \) (a vertical line upwards) and \( \frac{3\pi/4} \) (a line at 135 degrees).

3. **Plot the region**:
   - The region extends from the origin (r = 0) out to \( r = 2 \) along each of these angles.
   - You would shade the area enclosed between these two angles \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi/4} \), and between the radii \( r = 0 \) and \( r = 2 \).

4. **Shade the region**:
   - Start shading from the origin outward, following the angle \( \theta = \frac{\pi}{2} \) and extending to the angle \( \theta = \frac{3\pi/4} \).
   - Stop shading once you reach \( r = 2 \).

The shaded region should form a sector of a circle with the given radial bounds and angular bounds.

Would you like to see a diagram to visualize this region or any further details on this topic?

Here are five related questions to expand your understanding:
1. How do you convert polar coordinates to Cartesian coordinates?
2. What is the area of the shaded polar region?
3. How does the polar equation change if the radial range is extended?
4. What happens to the shaded region if the angle is decreased to \( \frac{\pi}{3} \)?
5. How would you graph the equation \( r = 1 + \cos(\theta) \) in polar coordinates?

**Tip:** When working with polar coordinates, remember that angles are always measured from the positive x-axis, moving counterclockwise.

Learn how to shade the polar region defined by 0 ≤ r ≤ 2 and π/2 ≤ θ ≤ 3π/4. Understand the concept of polar coordinates and how to plot and shade regions in polar coordinates. Suitable for university-level students studying mathematics.

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