Use a Riemann sum to find the area of the trapezoid bounded by the lines y = 0, x = 1, x = 2, and y = x. See Figure 1. %3D (2,2) y Cii) (0,0) X=1 X=2 メ

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Chapter1: Functions And Models
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**Finding the Area Using Riemann Sum**

**Objective**: Use a Riemann sum to find the area of the trapezoid bounded by the lines \( y = 0 \), \( x = 1 \), \( x = 2 \), and \( y = x \). See Figure 1 for reference.

**Explanation**:

In Figure 1, we have a graph that displays the equation \( y = x \), along with specific boundaries set by vertical and horizontal lines. These boundaries create a trapezoid, and our goal is to find its area.

**Key Points on the Graph**:
- The line \( y = x \) is drawn as a diagonal line from the origin \((0,0)\) through the graph.
- The trapezoid is bounded by:
  - \( y = 0 \) (the x-axis)
  - \( x = 1 \) (a vertical line at \( x = 1 \))
  - \( x = 2 \) (a vertical line at \( x = 2 \))
  - \( y = x \) (the diagonal line passing through points \((0,0)\) and \((2,2)\))

**Important Points Marked**:
- \((0,0)\): The origin
- \((1,1)\): Where \( x = 1 \) intersects \( y = x \)
- \((2,2)\): Where \( x = 2 \) intersects \( y = x \)
- Shaded Region: The area under the curve \( y = x \) from \( x = 1 \) to \( x = 2 \)

In using a Riemann sum, we are summing up the areas of rectangles (or trapezoids more commonly) under a curve to approximate the area. For more accurate results, the number of rectangles can be increased.

The shaded area represents the region we need to find the area for using the Riemann sum method. The limits for this calculation are from \( x = 1 \) to \( x = 2 \) under the line \( y = x \).

To calculate the area exactly, integrate \( y = x \) from \( x = 1 \) to \( x = 2 \):

\[ \text{Area} = \int_{1}^{2} x \, dx \]

Proceed with the integral calculation:
Transcribed Image Text:**Finding the Area Using Riemann Sum** **Objective**: Use a Riemann sum to find the area of the trapezoid bounded by the lines \( y = 0 \), \( x = 1 \), \( x = 2 \), and \( y = x \). See Figure 1 for reference. **Explanation**: In Figure 1, we have a graph that displays the equation \( y = x \), along with specific boundaries set by vertical and horizontal lines. These boundaries create a trapezoid, and our goal is to find its area. **Key Points on the Graph**: - The line \( y = x \) is drawn as a diagonal line from the origin \((0,0)\) through the graph. - The trapezoid is bounded by: - \( y = 0 \) (the x-axis) - \( x = 1 \) (a vertical line at \( x = 1 \)) - \( x = 2 \) (a vertical line at \( x = 2 \)) - \( y = x \) (the diagonal line passing through points \((0,0)\) and \((2,2)\)) **Important Points Marked**: - \((0,0)\): The origin - \((1,1)\): Where \( x = 1 \) intersects \( y = x \) - \((2,2)\): Where \( x = 2 \) intersects \( y = x \) - Shaded Region: The area under the curve \( y = x \) from \( x = 1 \) to \( x = 2 \) In using a Riemann sum, we are summing up the areas of rectangles (or trapezoids more commonly) under a curve to approximate the area. For more accurate results, the number of rectangles can be increased. The shaded area represents the region we need to find the area for using the Riemann sum method. The limits for this calculation are from \( x = 1 \) to \( x = 2 \) under the line \( y = x \). To calculate the area exactly, integrate \( y = x \) from \( x = 1 \) to \( x = 2 \): \[ \text{Area} = \int_{1}^{2} x \, dx \] Proceed with the integral calculation:
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