The *x* *y*-coordinate plane is given. There is a curve and a shaded region on the graph.

The curve begins at the point (0,1.0), moves down and to the right with increasing steepness, and ends at the approximate point (𝜋⁄2, 0.0).

The region is below the curve and above the *x*-axis between *x* = 0 and *x* = 𝜋⁄2. Determine the area of the given region.

The curve you described suggests a trigonometric function, likely \( y = \cos(x) \), as it starts at \( (0, 1) \) and reaches zero at \( x = \frac{\pi}{2} \). The area under this curve, from \( x = 0 \) to \( x = \frac{\pi}{2} \), can be found using integration.

To determine this area, we need to integrate \( \cos(x) \) with respect to \( x \) over the interval \( [0, \frac{\pi}{2}] \):

\[
\text{Area} = \int_0^{\frac{\pi}{2}} \cos(x) \, dx
\]

### Step-by-Step Solution:

1. **Set up the integral**:
   \[
   \int_0^{\frac{\pi}{2}} \cos(x) \, dx
   \]

2. **Find the antiderivative of** \( \cos(x) \):
   The antiderivative of \( \cos(x) \) is \( \sin(x) \).

3. **Evaluate the integral**:
   \[
   \int_0^{\frac{\pi}{2}} \cos(x) \, dx = \left[ \sin(x) \right]_0^{\frac{\pi}{2}}
   \]

4. **Plug in the bounds**:
   \[
   \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1
   \]

### Answer:
The area of the shaded region is \( \boxed{1} \).

Would you like more details or have any questions?

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Here are five additional questions to expand on this problem:

1. How would the area change if the curve extended from \( x = 0 \) to \( x = \pi \)?
2. What would the area be if the function were \( y = \sin(x) \) over the same interval?
3. How does this integral represent the concept of definite integration in calculus?
4. Can this problem be solved using Riemann sums instead of integration?
5. How would you find the area if the function had been shifted vertically?

**Tip:** Remember that the area under a curve between two points on the x-axis can always be found using definite integration.

The x y-coordinate plane is given. There is a curve and a shaded region on the graph. The curve begins at the point (0,1.0), moves down and to the right with increasing steepness, and ends at the approximate point (π⁄2, 0.0). The region is below the curve and above the x-axis between x = 0 and x = π⁄2. Determine the area of the given region.

Learn how to find the area of a shaded region under the curve y = cos(x) from x = 0 to x = π/2. This problem uses definite integration to determine the area between the curve and the x-axis, showcasing calculus concepts suitable for Grades 11-12.

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