Q.Find the area bounded by the curve $y=x^{2}+2$, the $x$-axis, and the ordinates $x=1$ and $x=3$.

The area under a non-negative curve between two vertical boundaries equals the definite integral of the curve's equation over that interval.

Since $y=x^{2}+2 \ge 0$ for every real $x$, the required area is exactly the definite integral:

Find an antiderivative and evaluate at the limits:

$F(3)=\frac{27}{3}+6=9+6=15, \qquad F(1)=\frac{1}{3}+2=\frac{7}{3}$ …

Evaluating $F(1)$ as a whole number by rounding $\frac{1}{3}+2$ to $2$ instead of keeping it as the exact fraction $\frac{7}{3}$ — carrying the fraction through to the fi …