Example 2. Calculate the area bounded by the curve $y=6 x-x^{2}-5$ and the $O x$ axis.
Solution. To determine the limits of integration, we find the points of intersection of the curve (parabola) with the $O x$ axis (Fig. 5.6). Solving the system $y=6 x-x^{2}-5, y=0$, we get $x_{1}=1, x_{2}=5$, hence, $a=1, b=5$.

Thus, the desired area is

$$
\begin{gathered}
S=\int_{1}^{5} y d x=\int_{1}^{5}\left(6 x-x^{2}-5\right) d x=6 \int_{1}^{5} x d x-\int_{1}^{5} x^{2} d x-5 \int_{1}^{5} d x= \\
=\left.6 \frac{x^{2}}{2}\right|_{1} ^{5}-\left.\frac{x^{3}}{3}\right|_{1} ^{5}-\left.5 x\right|_{1} ^{5}=3\left(5^{2}-1^{2}\right)-\frac{1}{3}\left(5^{3}-1^{3}\right)- \\
-5(5-1)=3 \cdot 24-\frac{1}{3} \cdot 124-5 \cdot 4=\frac{216-124-60}{3}= \\
=\frac{32}{3}=10 \frac{2}{3}
\end{gathered}
$$