定积分的本质
$$
\begin{aligned}
\int_a^b f(x)dx &= \lim_{\lambda \to 0} \sum_{i=1}^n f(\xi_i) \Delta x_i \\
\lambda &= \max{\Delta x_1,\Delta x_2,\Delta x_1,\cdots, \Delta x_n} \\
\end{aligned}
$$
教材参考章节
- 5.1 定积分的概念与性质
课后作业
- 试求函数$ y = \int_0^x \sin t dt $当 $x=0$ 及 $x=\frac{\pi}{4}$ 时的导数.
- 求$\frac{d}{dx}\int^{x^3}_{x^2}\frac{dt}{\sqrt{1+t^4}}$
- 求下列各定积分
$$
\begin{aligned}
&(1). \int_0^1\frac{dx}{\sqrt{4-x^2}} && (2). \int_0^{\frac{\pi}{4}} \tan ^2 \theta d\theta \\
&(3).\int_0^4 \frac{\ln x}{\sqrt{x}} dx &&(4).\int_1^e \sin(\ln x)dx\\
&(5). \int_{-1}^1 \frac{x dx}{\sqrt{5-4x}} &&(6). \int_{\frac{\sqrt{2}}{2}}^1 \frac{\sqrt{1-x^2}}{x^2} dx\\
\end{aligned}
$$