Use integration to find the area of the region enclosed by the graphs of  
y
=
√
x
+
2
 ,  
x
=
−
2
 , and  
y
=
3
 .


Round your result to 2 decimal place.

To find the area of the region enclosed by the graphs of the functions, we need to carefully set up an integral that represents the area.

### Problem:

The graphs are given as:
1. \( y = \sqrt{x} + 2 \)
2. \( x = -2 \)
3. \( y = 3 \)

The process involves finding the area between the curve \( y = \sqrt{x} + 2 \) and the horizontal line \( y = 3 \), bounded by \( x = -2 \).

### Steps:

1. **Find the intersection points:**

   The area is bound vertically between \( y = 3 \) and \( y = \sqrt{x} + 2 \), so we need to determine the limits of integration by solving for where these two curves intersect:
   \[
   y = \sqrt{x} + 2 = 3
   \]
   \[
   \sqrt{x} = 1 \quad \Rightarrow \quad x = 1
   \]
   So the curves intersect at \( x = 1 \).

2. **Set up the integral:**

   The integral will be set up from \( x = -2 \) to \( x = 1 \), and we are calculating the area between the line \( y = 3 \) and the curve \( y = \sqrt{x} + 2 \).

   The area is given by the integral of the difference between \( y = 3 \) and \( y = \sqrt{x} + 2 \):
   \[
   A = \int_{-2}^{1} \left( 3 - (\sqrt{x} + 2) \right) dx
   \]
   Simplifying the expression inside the integral:
   \[
   A = \int_{-2}^{1} (3 - \sqrt{x} - 2) dx = \int_{-2}^{1} (1 - \sqrt{x}) dx
   \]

3. **Evaluate the integral:**

   Now, we can compute the integral term by term:
   \[
   A = \int_{-2}^{1} 1 \, dx - \int_{-2}^{1} \sqrt{x} \, dx
   \]

Let's compute the value of the area.

Use integration to find the area of the region enclosed by the graphs of y = √x + 2, x = -2, and y = 3. Round your result to 2 decimal places.

This solution explains how to use integration to calculate the area enclosed by the graphs of y = √x + 2, x = -2, and y = 3. It involves setting up a definite integral, finding intersection points, and computing the area step by step. Suitable for advanced high school and college-level students.

Math Problem Statement

Use integration to find the area of the region enclosed by the graphs of
y

√ x + 2 ,
x

− 2 , and
y

Solution

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