To find the area of sector \( RST \) in circle \( S \), we can use the formula for the area of a sector, which is given by:
\[ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \( \theta \) is the angle of the sector in degrees, and \( r \) is the radius of the circle.
In this problem, we have:
- \( \theta = 134^\circ \)
- \( r = RS = 19 \)
Now we can substitute these values into the formula:
\[ \text{Area of sector RST} = \frac{134}{360} \times \pi \times 19^2 \]
Calculating \( 19^2 \):
\[ 19^2 = 361 \]
Now we substitute this value back into the area formula:
\[ \text{Area of sector RST} = \frac{134}{360} \times \pi \times 361 \]
Calculating \( \frac{134}{360} \):
\[ \frac{134}{360} \approx 0.3722 \]
Now we calculate the area:
\[ \text{Area of sector RST} \approx 0.3722 \times \pi \times 361 \]
Using \( \pi \approx 3.14159 \):
\[ \text{Area of sector RST} \approx 0.3722 \times 3.14159 \times 361 \]
Calculating \( 3.14159 \times 361 \):
\[ 3.14159 \times 361 \approx 1134.32299 \]
Now compute:
\[ \text{Area of sector RST} \approx 0.3722 \times 1134.32299 \approx 422.8759 \]
Rounding to the nearest hundredth:
\[ \text{Area of sector RST} \approx 422.88 \]
Thus, the area of sector \( RST \) is approximately \( \boxed{422.88} \) square units.