Find the area under the given curve over the indicated interval. \[ y=x^{2}+x+2 ;[4,6] \]

Step 1 ：We are given the function \(y = x^{2} + x + 2\) and we need to find the area under the curve from \(x = 4\) to \(x = 6\).

Step 2 ：The area under a curve from a to b is given by the definite integral of the function from a to b.

Step 3 ：So, we need to find the definite integral of the function \(y = x^{2} + x + 2\) from 4 to 6.

Step 4 ：By calculating the definite integral, we find that the area under the curve over the indicated interval is \(\frac{194}{3}\).

Step 5 ：Final Answer: The area under the curve over the indicated interval is \(\boxed{\frac{194}{3}}\).