问题 问答题

(x≥-1),求曲线y=f(x)与x轴所围成的封闭图形的面积.

答案

参考答案:因为t|t|为奇函数,可知其原函数
[*]
为偶函数,而f(-1)=0,f(1)=0 (因为t|t|是奇函数),即y=f(x)与x轴有交点(-1,0),(1,0).又由f’(x)=x|x|,可知x<0时f’(x)<0,故f(x)单调下降,从而f(x)<f(-1)=0(-1<x≤0).
当x>0时,f’(x)=x|x|>0,故x>0时f(x)单调上升,y=f(x)与x轴有唯一交点(1,0),因此y=f(x)与x轴交点仅有两个.
所以封闭曲线所围成的面积
[*]
当x<0时,[*],故
[*]

解析:

[分析]: 由于被积函数t|t|是奇函数,从而可知f(x)是偶函数,再由f’(x)的符号,可求出y=f(x)在x轴的两个交点,最后由定积分的几何意义,可求出所论图形的面积.

阅读理解

阅读理解

                                                             Body and Food

      Your body, which has close relations with the food you eat, is the most important thing you own, so

it needs proper treatment and proper nourishment (营养).The old saying "An apple a day keeps the

doctor away"is not as silly as some people think. The body needs fruit and vegetables because they

contain vitamin C. Many people take extra vitamins in pill form, believing that these will make them

healthy.

     But a good diet is made up of nourishing food and this gives all the vitamins you need. The body

doesn't need or use extra vitamins, so why waste money on them?

     In the modern western world, many people are too busy to bother about eating properly. They throw

anything into their stomachs, eating hurriedly and carelessly. The list of illnesses caused or made worse

by bad eating habits is frightening,

1. "Your body has close relations with the food you eat." It really means that ______.

A. all kinds of food you eat can be made into your body

B. your body is made up of the food you eat

C. what you eat has great effect on your health

D. the more you eat, the fitter you will feel

2. The old saying referred to in the, passage tells us that ______.

A. eating apples regularly does lots of good to our health

B. the apple is the best among all kinds of fruits

C. apples can take the place of doctors

D. an apple is a sure cure for illness

3. In the second paragraph, the writer tries to let us know ______.

A. our bodies need food or we can't live

B. often eating apples is a good habit

C. taking extra vitamin pills is completely useless

D. a good diet is of great importance for our health

4. In the modern western countries ______.

A. people don't want to pay more attention to their eating

B. lots of people's illnesses are caused or made worse by bad eating habits

C. people throw everything into their stomachs without chewing

D. people are only too busy to cook meals for themselves

5. From the passage we can draw a conclusion that if we want to keep healthy,we should ______.

A. only eat an apple a day

B. eat properly

C. take as many vitamin pills as possible

D. throw something into our stomachs slowly and carefully

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