find the area between the curves y=\frac{x}{2} and y=\sqrt{x+2} using integration

To find the area between the curves \( y = \frac{x}{2} \) and \( y = \sqrt{x+2} \) using integration, we need to follow these steps:

### Step 1: Determine the points of intersection.
We first need to find where the two curves intersect by solving the equation:

\[
\frac{x}{2} = \sqrt{x+2}
\]

Square both sides to remove the square root:

\[
\left( \frac{x}{2} \right)^2 = ( \sqrt{x+2} )^2
\]
\[
\frac{x^2}{4} = x + 2
\]

Multiply both sides by 4 to eliminate the fraction:

\[
x^2 = 4(x + 2)
\]
\[
x^2 = 4x + 8
\]

Rearrange the equation:

\[
x^2 - 4x - 8 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-8)}}{2(1)}
\]
\[
x = \frac{4 \pm \sqrt{16 + 32}}{2}
\]
\[
x = \frac{4 \pm \sqrt{48}}{2}
\]
\[
x = \frac{4 \pm 4\sqrt{3}}{2}
\]
\[
x = 2 \pm 2\sqrt{3}
\]

So, the two intersection points are \( x = 2 + 2\sqrt{3} \) and \( x = 2 - 2\sqrt{3} \). However, we are only interested in the real, positive intersection point since we're working with the curves over real values of \( x \). Thus, the relevant intersection point is:

\[
x = 2 + 2\sqrt{3}
\]



Find the area between the curves y=\frac{x}{2} and y=\sqrt{x+2} using integration

Learn how to find the area between the curves y=\frac{x}{2} and y=\sqrt{x+2} through integration. This solution involves solving quadratic equations and applying the fundamental theorem of calculus to calculate the area. Ideal for students in Grades 11-12.

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