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2018年优课系列高中数学北师大版选修2-2 4.1.2定积分 课件(12张)

1、求曲边梯形面积和变速直线运动 路程的步骤是什么? 2、求曲边梯形面积的公式是什么? 3、求变速直线运动路程的公式是什么?
4、它们的共同特征是什么?

§1.5 定 积 分 --§1.5.3定积分的概念

? ? n
i ?1

f (?i )?x ?

n i ?1

f

(?i

)

?

b

? n

a

如果当n?∞时,上述和式无限接近某个常数,

?? 这个常数为函数f(x)在区间[a, b]上的定积分,

记作

? ? ? b f ( x ) d x , 即 b f ( x ) d x ? l n f i ( i ) ? ? x i 。 m

a

a ? 0 i ? 1

? ? 即

b a

f

( x)dx

?

lim
n??

n i?1

b?a n

?

f

(?i )

积分上限

[a, b] 叫做积分区间







? ? b f (x)dx ? lim

n

b ? a f (?

)

a

n n?? i?1

i



积分下限

被 积

被 积

积 分











合作探究

(1)定积分的结果是一个 数值

(2)定积分的值只与被积函数和积分区 间有关,而与积分变量用什么字母表 示 无关 , 即

? ? b f (x)dx =

b
f (t)dt

a

a

如何用定积分表示抛物线 y ? x2 、
直线 x ?1 和 x 轴所围成的曲边梯形
的面积。



y





y ? x2

O

1x

定积分的几何意义( f (x) ? 0 )

设阴影部分面积为S

b
?a f (x)dx

表示由直线 x ? a,

x ? b (a ? b), y ? 0

和曲线 y ? f (x) 所

b
?a f (x)dx ? S

围成的的曲边梯形 的面积

合作探究
如何用定积分表示图中蓝色部分的面积?
y?f (x) y

Oa

y ? g?x?

b

b

?a f (x)dx ? ?a g(x)dx

bx

用定积分表示下列图中阴影部分的面积

y

y ? 2x

y



y ? sin x





01 x

0 1 3?

x

4



1
?0 2xdx

3?
? 4 sin xdx 1

? 例⒈利用定积分的定义,计算 1 x3dx 的值 0

? 注: n i3 ? 13 ? 23 ? 33 ? ??? ? n3 ? 1 n2 (n ?1)2

i?1

4

小结
1、通过本节课的学习,你学到 了哪些知识? 2、本节课用到了哪些思想方法?


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