定积分
什么是定积分?
- 这就是所谓的基本定理的微积分
- 一个和b被称为极限
- 一个是下限
- b是上限
" class="Wirisformula" role="math" alt="空格f左括号x右括号" style="vertical-align:-6px;height:22px;width:35px">是被积函数f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格F左括号x右括号" style="vertical-align:-6px;height:22px;width:37px">是一个不定积分的F ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f左括号x右括号" style="vertical-align:-6px;height:22px;width:35px">f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的常数的集成(“+c)中不需要明确的集成
- " +c会出现在两者的旁边F (一个)和F (b)
- 减法意味着“+”c”的取消
我如何找到定积分分析(手动)?
表示法:使用方括号[],限制放在括号的末尾
考试技巧
- 如果问题没有说明你可以使用你的GDC,那么你必须清楚地展示你的所有工作,但是如果你在考试中使用了你的GDC,那么使用它来检查你的答案是一个很好的做法
工作的例子
定积分的性质
微积分基本定理
- 在形式上,
" class="Wirisformula" role="math" alt="空格f左括号x右括号" style="vertical-align:-6px;height:22px;width:35px" loading="lazy">是连续在这段时间f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格a小于等于x小于等于b" style="vertical-align:-6px;height:22px;width:66px" loading="lazy">a ≤ x ≤ b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格F左括号x右括号" style="vertical-align:-6px;height:22px;width:37px" loading="lazy">是一个不定积分的F ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格f左括号x右括号" style="vertical-align:-6px;height:22px;width:35px" loading="lazy">f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
定积分的性质是什么?
- 其中有些已经遇到过,有些似乎是显而易见的……
- 采取常数积分外的因子
" class="Wirisformula" role="math" alt="积分下标a上标b k f左括号x右括号空格d x等于k积分下标a上标b f左括号x右括号空格d x" style="vertical-align:-22px;height:49px;width:196px" loading="lazy">在哪里∫ a b k f ( x ) d x = k ∫ a b f ( x ) d x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空间k" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">是常数k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 当涉及小数和/或负值时很有用
- 逐项积分
" class="Wirisformula" role="math" alt="空格积分下标a上标b左方括号直f左括号x右括号加上直g左括号x右括号右方括号空格直dx等于积分下标a上标b直f左括号x右括号空格直dx加上积分下标a上标b直g左括号x右括号空格直dx" style="vertical-align:-22px;height:49px;width:337px" loading="lazy">∫ a b [ f ( x ) + g ( x ) ] d x = ∫ a b f ( x ) d x + ∫ a b g ( x ) d x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 上述方法也适用于项/函数的减法
- 上下限相等
" class="Wirisformula" role="math" alt="积分下标一个上标一个f左括号x右括号空间dx = 0" style="vertical-align:-22px;height:49px;width:105px" loading="lazy">∫ a a f ( x ) d x = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 在求值时,这将是一个值,减去自己!
- 交换限制会得到相同但负的结果
" class="Wirisformula" role="math" alt="积分下标a上标b f左括号x右括号空格d x等于负的积分下标b上标a f左括号x右括号空格d x" style="vertical-align:-22px;height:49px;width:192px" loading="lazy">∫ a b f ( x ) d x = - ∫ b a f ( x ) d x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 比较8减5和5减8……
- 分割间隔
" class="Wirisformula" role="math" alt="空格积分下标a上标b直f左括号x右括号空格直dx等于积分下标a上标c直f左括号x右括号空格直dx加上积分下标c上标b直f左括号x右括号空格直dx" style="vertical-align:-22px;height:49px;width:275px" loading="lazy">在哪里∫ a b f ( x ) d x = ∫ a c f ( x ) d x + ∫ c b f ( x ) d x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格a小于等于c小于等于b" style="vertical-align:-6px;height:22px;width:65px" loading="lazy">a ≤ c ≤ b {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 这对于多曲线下的区域或曲线下的区域特别有用
" class="Wirisformula" role="math" alt="空间x" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">设在x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 水平翻译
" class="Wirisformula" role="math" alt="空格积分下标a上标b直f左括号x右括号空格直dx等于积分下标a - k末端下标b - k末端上标直f左括号x加上k右括号空格直dx" style="vertical-align:-22px;height:49px;width:227px" loading="lazy">在哪里∫ a b f ( x ) d x = ∫ a - k b - k f ( x + k ) d x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空间k" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">是常数k {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 的图形
" class="Wirisformula" role="math" alt="空格y等于f左括号x正负k右括号" style="vertical-align:-6px;height:22px;width:88px" loading="lazy">这幅图的水平平移吗y = f ( x ± k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="空格y等于f左括号x右括号" style="vertical-align:-6px;height:22px;width:62px" loading="lazy">y = f ( x ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
( " class="Wirisformula" role="math" alt="f左括号x + k右括号" style="vertical-align:-6px;height:22px;width:58px" loading="lazy">翻译,f ( x + k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="f左括号x - k右括号" style="vertical-align:-6px;height:22px;width:58px" loading="lazy">翻译)f ( x - k ) {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 采取常数积分外的因子
考试技巧
- 学习定积分的性质有助于在考试中节省时间
工作的例子
众所周知,