Answer:
To find the area under the curve represented by the equation \(x = y^2 + 2\), you'll need to determine the limits of integration and set up a definite integral. Since you haven't specified the range over which you want to find the area, I'll assume you want to find the area between two values of \(y\), say \(y = a\) and \(y = b\).
The area under the curve between \(y = a\) and \(y = b\) can be calculated using the following definite integral:
\[A = \int_{a}^{b} (x) \, dy\]
Now, you need to express \(x\) in terms of \(y\) using the equation \(x = y^2 + 2\):
\[A = \int_{a}^{b} (y^2 + 2) \, dy\]
Next, you'll integrate with respect to \(y\) from \(a\) to \(b\) to find the area:
\[A = \left[\frac{1}{3}y^3 + 2y\right]_{a}^{b}\]
Now, you can calculate the area by substituting \(b\) and \(a\) into the equation and subtracting the results:
\[A = \left[\frac{1}{3}b^3 + 2b\right] - \left[\frac{1}{3}a^3 + 2a\right]\]
This expression will give you the area under the curve \(x = y^2 + 2\) between the values of \(y = a\) and \(y = b\). If you have specific values for \(a\) and \(b\), you can substitute them into this expression to calculate the area.
Explanation: