Applications of Integration
AP Calculus AB Topic 8 6:04 English narration · English + 中文 subtitles burned in
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What is the average value of a function?
一个函数的平均值是什么？
Not the average of a few points — the average across a whole interval.
不是几个点的平均，而是整个区间上的平均。
Picture the area under the curve.
想象曲线下方的面积。
Now flatten it into a rectangle of the same width, holding exactly the same area.
现在把它压平成一个同样宽度的矩形，面积保持完全相同。
The height of that rectangle is the average value.
那个矩形的高，就是平均值。
So the average value is the integral, divided by the length of the interval.
所以平均值就是积分，除以区间的长度。
Today we put integrals to work — average value, motion, areas between curves, and the volumes of solids.
今天我们让积分派上用场——平均值、运动、曲线之间的面积，以及立体的体积。
Let's begin.
让我们开始吧。
Here is the formula.
这就是公式。
The average value of f on the interval from a to b is one over b minus a, times the integral of f.
f 在从 a 到 b 区间上的平均值，等于 b 减 a 分之一，乘以 f 的积分。
For x squared on the interval zero to three, the integral comes to nine, and dividing by three gives an average value of three.
对区间零到三上的 x 平方，积分得到九，再除以三，得到平均值三。
Be careful: this is not the average rate of change, which divides the change in f by the interval.
要小心：这不是平均变化率，那个是用 f 的变化量除以区间长度。
The exam tests that difference.
考试就爱考这个区别。
Do not confuse the average value with a Riemann-sum average of a few values — the exam distinguishes these carefully.
不要把平均值和几个点的黎曼和平均搞混——考试会仔细区分这两者。
Report units in context.
还要在情境里报单位。
If a rate is in vehicles per hour, the average value is in vehicles per hour too.
如果变化率的单位是辆每小时，平均值的单位也是辆每小时。
Integration reverses the motion links from earlier.
积分把前面讲过的运动关系反了过来。
Over a time interval, the integral of velocity gives the displacement — the net change in position.
在一段时间区间上，速度的积分给出位移—— 也就是位置的净变化。
But if you want the total distance travelled, integrate the speed — the absolute value of velocity — so a change of direction adds instead of cancelling.
但如果你想要走过的总路程，就要对速率积分——也就是速度的绝对值—— 这样方向的改变就是相加而不是抵消。
And to get the position at a later time, start from the known position and add the integral of velocity.
而要得到之后某时刻的位置， 就从已知位置出发，加上速度的积分。
That is one case of the net change theorem, and it is everywhere on the exam.
这只是净变化定理的一个特例，而这个定理在考试里无处不在。
The integral of a rate, over an interval, is the net change in the quantity.
一个变化率在区间上的积分，就是那个量的净变化。
So the final amount equals the initial amount, plus the integral of the rate.
所以最终量等于初始量，加上变化率的积分。
How much water is in the tank at time five?
时刻五时水箱里有多少水？
Start with what was already there, and add the integral of inflow minus outflow.
先取原本已有的量，再加上流入减流出的积分。
Watch the signs, and watch the units.
注意符号，也注意单位。
A function defined as an integral accumulates a rate of change.
一个被定义为积分的函数，就是在积累变化率。
Accumulation functions and definite integrals in applied contexts are the same idea: the definite integral of a rate over an interval is the net change of the quantity.
应用情境里的积累函数和定积分，其实是同一件事： 变化率在区间上的定积分，就是那个量的净变化。
Many multi-part free-response questions are built entirely on this.
很多多小题的自由作答题完全建立在这个想法上。
Often they ask you to write, but do not evaluate, an integral expression that gives the total.
它们常常让你写出、但不要计算、一个给出总量的积分表达式。
The integral is already the answer they want.
积分本身就是它们要的答案。
To find the area between two curves, integrate top minus bottom.
要求两条曲线之间的面积，就积分"上减下"。
Find where they intersect first — those crossings give you the limits.
先找出它们的交点——那些交点给出积分限。
For the line y equals x and the parabola y equals x squared, they cross at zero and one, the line is on top, and the area works out to one sixth.
对直线 y 等于 x 和抛物线 y 等于 x 平方，它们在零和一处相交，直线在上方， 面积算出来是六分之一。
If the region is easier in horizontal slices, integrate right minus left with respect to y instead.
如果这个区域用水平的薄片更好处理， 那就改成对 y 积分"右减左"。
And if the curves cross more than twice, split into a sum of integrals at each crossing — because the top curve changes.
而如果曲线相交超过两次，就在每个交点处拆成积分之和—— 因为上方的曲线换了。
Now volumes.
现在讲体积。
If a solid has a known cross section perpendicular to an axis, its volume is simply the integral of the cross-sectional area.
如果一个立体在垂直于某轴的方向上有已知的横截面， 那么它的体积就是横截面面积的积分。
Slice it thin, find the area of one slice, then add them all up.
把它切成薄片，求出一片的面积，再把它们全加起来。
For square cross sections, the area is the slice length squared.
对正方形横截面，面积是切片长度的平方。
For an equilateral triangle it is root three over four, times that square.
对等边三角形，是四分之根号三乘以那个平方。
For a semicircle, pi over eight times the square.
对半圆，是八分之派乘以那个平方。
Same method — just a different area formula.
方法相同——只是面积公式不同。
Revolve a region around an axis, and it sweeps out a solid of revolution.
让一个区域绕轴旋转，它就扫出一个旋转体。
If the region touches the axis, every slice is a solid disc, with radius equal to the function value.
如果这个区域与轴相接， 那么每一个切片都是一个实心圆盘，半径等于函数值。
The volume of one thin disc is pi, times the radius squared, times its thickness — so the total volume is pi, times the integral of the radius squared.
一个薄圆盘的体积是派， 乘以半径的平方，再乘以它的厚度——所以总体积就是派，乘以半径平方的积分。
Let's use it.
我们来用一用。
Revolve the region under the square root of x, from zero to four, about the x-axis.
把 x 的平方根 下方、从零到四的区域，绕 x 轴旋转。
The radius of each disc is the square root of x — so the radius squared is just x.
每个圆盘的半径是 x 的平方根——所以半径的平方就是 x。
The volume is pi, times the integral of x from zero to four, which is pi times eight.
体积是派， 乘以 x 从零到四的积分，也就是派乘以八。
So the volume is eight pi.
所以体积是八派。
Around any horizontal or vertical line, the radius is the distance from the curve to that line.
绕任何水平或竖直线旋转时，半径就是从曲线到那条线的距离。
Set up the radius carefully, then use pi times the integral of radius squared.
把半径设对， 然后用派乘以半径平方的积分。
Revolving about y equals negative two is a very common exam variation.
绕 y 等于负二旋转是很常见的考法。
Measure both distances to that line, not to an axis.
两个距离都要量到那条线，而不是量到坐标轴。
But what if the region does not touch the axis?
但如果这个区域并不与轴相接呢？
Then each slice has a hole in the middle — a washer.
那么每一个切片中间就有一个洞——这叫垫圈。
R is the distance from the axis to the farther boundary and r to the nearer one.
R 是从轴到较远边界的距离，r 是到较近边界的距离。
Take the outer radius, square it, and subtract the square of the inner radius.
取外半径，平方，再减去内半径的平方。
The volume is pi, times the integral of that difference.
体积是派，乘以这个差的积分。
Both radii are distances measured from the axis of rotation — and when you revolve about a line that is not an axis, measure both distances to that line.
两个半径都是从旋转轴量起的距离——而当你绕一条不是坐标轴的直线旋转时， 两个距离都要量到那条线。
Before you go, three marks to keep.
结束之前，三个要守住的分。
First, area between curves is always top minus bottom — find the intersections for your limits.
第一，曲线之间的面积永远是上减下——先求交点作为积分限。
Second, displacement uses velocity, but total distance uses the absolute value of velocity — never mix them.
第二，位移用速度，但总路程用速度的绝对值——绝不要混淆。
Third, for a washer, subtract the squares of the radii, not the square of the difference.
第三，对垫圈，是把两个半径分别平方再相减，而不是把差再平方。
And remember: the average value is the integral divided by the interval length.
还要记住：平均值是积分除以区间长度。
This topic is yours.
这个专题就是你的了。