Example 6. Calculate the area bounded by the parabola $x=8 y-y^{2}-7$ and the $O y$ axis.
Solution. To determine the limits of integration, we find the points of intersection of the curve with the $O y$ axis.

Solving the system of equations $x=8 y-y^{2}-7, x=0$, we get $\quad y_{1}=1, \quad y_{2}=7$

(Fig. 5.10). By formula (5.16) we find

$$
\begin{aligned}
& S=\int_{1}^{7}\left(8 y-y^{2}-7\right) d y=8 \int_{1}^{7} y d y-\int_{1}^{7} y^{2} d y-7 \int_{1}^{7} d y= \\
& =\left.8 \cdot \frac{y^{2}}{2}\right|_{1} ^{7}-\left.\frac{y^{3}}{3}\right|_{1} ^{7}-\left.7 y\right|_{1} ^{7}=4(49-1)-\frac{1}{3}(343-1)- \\
& -7(7-1)=192-114-42=36
\end{aligned}
$$