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人教A版高中数学选修2-2课件0225高二(1.7.1定积分在几何中的应用)_图文

高中数学课件
(金戈铁骑 整理制作)

1.7 定积分的简单应用
1.7.1 定积分在几何中的应用

问题提出

t

p

?

? ??

1 2

? ??

5730

b
1分.别定是积什分么òa f (x )d的x 含义及其几何意义

ò ? b
f (x )dx
a

=

lim
n

n i= 1

b

n

a

f

(xi )

y y=f(x)

b
òa f (x)dx

Oa

bx

2.微积分基本定理是什么? 如果f(x)是区间[a,b]上的连续函数,

并且 F ?(x ) = f (x ) ,则

蝌b

b

. f (x )dx = F ?(x )dx = F (b) - F (a)

a

a

3.用定积分可以表示曲边梯形的面积, 微积分基本定理为定积分的计算提供了 一种有效的方法,二者强强联合,可以 解决平面几何中曲边图形的面积问题.

探究(一):曲线y2=x与y=x2所围成图 形的面积

思考1:曲线y2=x与y=x2所围成的图形 是什么?其交点坐标是什么?

y=x2

y

y2=x

1

(1,1)

O

1

x

(0,0)

思考2:如何将该图形的面积转化为曲边

梯形的面积?

y

y=x2

y2=x

1C B

DA

O

1

x

S=S曲边梯形OABC-S曲边梯形OADC.

思考3:该图形的面积用定积分怎样表示?

y=x2

y

1C B

DA

O

1

y2=x x

蝌1

1

S=

xdx - x 2dx

0

0

思考4:利用微积分基本定理计算,该图

形的面积等于多少?

y

y=x2

y2=x

1C B

DA

O

1

x

S

=

2

x

3 2

3

|10

-

1 3

x3

|10 =

1 3

探究(二):直线y=x-4与曲线y = 2x 及x轴所围成图形的面积

思考1:直线y=x-4与曲线 y = 2x 及

x轴所围成的图形是什么?各顶点的坐标

是什么?

y

y=x-4

4

y = 2x
(8,4)

(0,0)
O

4 8x
(4,0)

思考2:如何将该图形的面积转化为曲边

梯形的面积? y

y=x-4

4 C

y = 2x
B

A

O

D4 8 x

S=S曲边梯形OABC-S三角形ABD.

思考3:该图形的面积用定积分怎样表

示?

y

y=x-4

4 C

y = 2x
B

A

O

D4 8 x

蝌8

8

S = 2xdx - (x - 4)dx

0

4

思考4:利用微积分基本定理计算,该图

形的面积等于多少?
y

y=x-4

4 C

y = 2x
B

A

O

D4 8 x

S

=

22 3

3
x2

|80

-

1 创4 2

4 = 40 3

理论迁移
例1 计算由直线y=2-x,y = - 1 x
3

和曲线y = x 所围成的平面图形的面积.

y

y=2-x B
1

O

1

-1

y= x

23 x
A

S = 13 6

例2 如图,直线y=kx将抛物线 y=x-x2与x轴所围成的平面图形分成 面积相等的两部分,求实数k的值.

1 k = 1- 3
2

y y=x-x2 y=kx

1

O

1-k

x

小结作业
1.定积分在几何中的应用,主要用于求 平面曲边图形的面积.解题时,一般先要 画出草图,再根据图形确定被积函数以 及积分的上、下限. 2.定积分只能用于求曲边梯形的面积, 对于非规则曲边梯形,一般要将其分割 或补形为规则曲边梯形,再利用定积分 的和与差求面积.对于分割或补形中的多 边形的面积,可直接利用相关面积公式 求解.

3.位于x轴下方的曲边梯形的面积,等于

相应定积分的相反数.一般地,设由直线

x=a,x=b(a<b),y=0和曲线y=f(x)

所围成的曲边梯形的面积为S,则.

b
S = òa | f (x ) | dx

y y=|f(x)|

Oa

bx

y=f(x)

作业: P58练习:(1),(2). P60习题1.7B组:1,2,3.


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