https://file.aiquickdraw.com/mathbot/file/1728501988039i8bceocf.jpg

To graph this, we are dealing with a curve, axes, and a vertical line. Let's break it down:

### 1. **Equation of the curve:**
   The given equation is:
   \[
   y = x^2 + 1
   \]
   This is a parabola that opens upwards with its vertex shifted one unit upwards (because of the "+1" term). The vertex is at (0,1).

### 2. **Boundaries:**
   - The region is bounded by the **x-axis** (\(y = 0\)) and the **y-axis** (\(x = 0\)).
   - There's also a boundary at the vertical line **x = 2**.

### 3. **Graphing steps:**
   1. **Plot the curve \(y = x^2 + 1\)**:
      - At \(x = 0\), \(y = 1\).
      - At \(x = 1\), \(y = 2\) (since \(1^2 + 1 = 2\)).
      - At \(x = 2\), \(y = 5\) (since \(2^2 + 1 = 5\)).

   2. **Plot the axes**:
      - The x-axis is the horizontal line where \(y = 0\).
      - The y-axis is the vertical line where \(x = 0\).

   3. **Draw the line \(x = 2\)**:
      - This is a vertical line passing through \(x = 2\) that intersects the curve at the point (2, 5).

### 4. **Sketch the bounded region**:
   The region is bounded by:
   - The **curve** \(y = x^2 + 1\) from \(x = 0\) to \(x = 2\),
   - The **x-axis** from \(x = 0\) to \(x = 2\),
   - The **y-axis** at \(x = 0\), and
   - The vertical line \(x = 2\).

The region in the first quadrant (QI) is enclosed by these elements.

Would you like me to generate a graph for this description or explain more details? Here are some questions to expand on this:

1. How can you find the area under a curve using integration?
2. What is the significance of the limits \(x = 0\) and \(x = 2\)?
3. How do you set up an integral to calculate the area under a parabola?
4. What happens if the line \(x = 2\) is moved further right, say to \(x = 3\)?
5. How would the area change if the equation was \(y = x^2\) (without the +1)?

**Tip:** To find the area under a curve, we often use definite integrals with the limits of the given region.

Find the area of the region in QI bounded by the curve y = x^2 + 1, both axes, and the line x=2.

Explore how to find the area of the region bounded by the curve y = x^2 + 1, both axes, and the line x=2 in the first quadrant. Includes step-by-step integration and sketching details. Ideal for high school students studying calculus and areas under curves.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation