337. $y=x^{2}+1, x=-2, x=2$ and $y=0$ (Fig. 185).
Solution. \( S = \int_{-2}^{2} \left(x^{2} + 1\right) dx = 2 \int_{0}^{2} \left(x^{2} + 1\right) dx = 2 \left( \frac{x^{3}}{3} + x \right) \bigg|_{\sigma}^{2} = 9 \frac{1}{3} \) (sq. units).

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-331.jpg?height=311&width=483&top_left_y=1073&top_left_x=166)

Fig. 186

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-331.jpg?height=338&width=344&top_left_y=1046&top_left_x=716)

Fig. 187

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-331.jpg?height=365&width=252&top_left_y=1448&top_left_x=270)

Fig. 188

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-331.jpg?height=360&width=390&top_left_y=1443&top_left_x=582)

Fig. 189