曲线 C 由参数方程定义： $x = \theta^2 - 1,\quad y = \tan\frac{\theta}{2},\quad 0 \le \theta < \pi$。 求由 C、y 轴和 x 轴所围成且满足 $x \le 0$ 的区域的面积。将你的答案精确到 4 位小数。

1. Determine the limits of $\theta$ corresponding to the region bounded by the coordinate axes for $x \le 0$.

The curve intersects the x-axis ($y = 0$):
$\tan\left(\frac{\theta}{2}\right) = 0 \implies \theta = 0$
At $\theta = 0$, $x = -1$.

The curve intersects the y-axis ($x = 0$):
$\theta^2 - 1 = 0 \implies \theta = 1$ (since $0 \le \theta < \pi$)

The region bounded by the curve and the axes for $x \le 0$ corresponds to $0 \le \theta \le 1$.

2. Formulate the definite integral for the area under the curve.

The area bounded by the curve, the x-axis, and the y-axis is given by $\int_{x_1}^{x_2} y \, dx$.
Using the parametric equations, we find $\frac{dx}{d\theta}$:
$x = \theta^2 - 1 \implies \frac{dx}{d\theta} = 2\theta \implies dx = 2\theta \, d\theta$

Substituting $y$ and $dx$ into the integral over the interval $0 \le \theta \le 1$:
$\text{Area} = \int_{0}^{1} \tan\left(\frac{\theta}{2}\right) (2\theta) \, d\theta$

3. Evaluate the integral to 4 decimal places using a graphing calculator.

Using a graphing calculator (in radian mode):
$\text{Area} = \int_{0}^{1} 2\theta \tan\left(\frac{\theta}{2}\right) \, d\theta \approx 0.35129...$

Rounding to 4 decimal places:
$\text{Area} = 0.3513$