We are given a sector of a circle and are asked to calculate the area of the shaded region. Here's how we can solve it step by step:

1. Understand the geometry:

• The circle is centered at the origin $O(0,0)$ with point $B(2, 0)$ on the x-axis, indicating that the radius of the circle is $2$ units.
• Point $A \left( \sqrt{2}, \sqrt{2} \right)$ forms the vertex of the shaded sector.

2. Calculate the angle of the sector:

• The angle between the points $B(2, 0)$ and $A(\sqrt{2}, \sqrt{2})$ can be found using the dot product formula for vectors.

• The angle $\theta$ between two vectors $\vec{OB}$ and $\vec{OA}$ is given by: $\cos(\theta) = \frac{\vec{OB} \cdot \vec{OA}}{\|OB\| \|OA\|}$ where:

  • $\vec{OB} = (2, 0)$,
  • $\vec{OA} = (\sqrt{2}, \sqrt{2})$,
  • $\|OB\| = 2$ and $\|OA\| = 2$.

  Calculate the dot product: $\vec{OB} \cdot \vec{OA} = (2)(\sqrt{2}) + (0)(\sqrt{2}) = 2\sqrt{2}.$

  Now, substitute into the formula for the angle: $\cos(\theta) = \frac{2\sqrt{2}}{2 \times 2} = \frac{\sqrt{2}}{2}.$

  Thus, $\theta = 45^\circ$ or $\frac{\pi}{4}$ radians.

3. Calculate the area of the sector:

The formula for the area of a sector is: $\text{Area of sector} = \frac{1}{2} r^2 \theta.$ Here, $r = 2$ and $\theta = \frac{\pi}{4}$.

Substituting the values: $\text{Area of sector} = \frac{1}{2} \times 2^2 \times \frac{\pi}{4} = \frac{1}{2} \times 4 \times \frac{\pi}{4} = \frac{4\pi}{8} = \frac{\pi}{2}.$

Thus, the area of the shaded region is $\frac{\pi}{2}$.

Answer:

The correct answer is E. $\frac{\pi}{2}$.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How is the dot product of two vectors calculated geometrically?
2. What other methods can be used to find the angle between two vectors?
3. How does the area of a sector change with different angles or radii?
4. What are the practical applications of sector area calculations in real life?
5. How does the formula for the sector area differ from the area of a full circle?

Tip: Always ensure the angle used in sector area calculations is in radians, as the formula $\frac{1}{2} r^2 \theta$ assumes this!