Find the area under the curve Y=x+1, Y=0, X=2, X=6

First, set up the definite integral:

\int_{2}^{6} (x+1) \, dx

Calculate the antiderivative:

\int (x+1) \, dx = \frac{x^2}{2} + x

Evaluate the antiderivative at the upper and lower limits of the integral:

\left[ \frac{x^2}{2} + x \right]_{2}^{6}

Substitute the limits:

\left( \frac{6^2}{2} + 6 \right) - \left( \frac{2^2}{2} + 2 \right)

Simplify:

\left( 18 + 6 \right) - \left( 2 + 2 \right)

24 - 4 = 20

So, the area under the curve is:

20