In the previous chapter, we approximated the net area under the curve \(y=f(x)\) by dividing the interval of \(x\) into small subintervals and forming the Riemann sums. As the maximum length of the subintervals goes to zero (and the number of subintervals goes to infinity), these sums approach a limit that is equal to the exact area. Next, we learned how to calculate this limit using the Fundamental Theorem of Calculus. Actually the previous chapter may be summarized by the following equation \[\begin{aligned} \lim_{\text{max}\Delta_{k}x\to0}\sum_{k=1}^{n}f(x_{k}^{*})\Delta_{k}x & =\int_{a}^{b}f(x)dx\\ & =F(x)\bigg|_{x=a}^{x=b}=F(b)-F(a),\end{aligned}\] where \(F\) is an antiderivative (also called an integral function) of \(f\).

Many other quantities such as the volume and surface of a solid, the length of a curve, and the work done by a variable force can be calculated by essentially the same process. We divide the interval of the independent variable into small pieces, form the corresponding sums to approximate these quantities, and then take the limit of these sums. This limit is equal to the exact value of the quantity.

Once the limit-of-sums process is understood, we do not have to think about every single detail over and over again for each quantity that we need to calculate. Instead, we can take an intuitive shortcut to skip the details and develop the formulas much faster.

Once again let’s look at the formula \[\text{Area }A=\int_{a}^{b}f(x)dx\quad(\text{or }A=\int_{a}^{b}ydx),\] since \(y=f(x)\). As shown in Figure 1, we can imagine that \(f(x)dx\) (or \(ydx\)) is the area of a rectangle of base \(dx\) and height \(f(x)\). This rectangle is located at a position specified by a value of \(x\) between \(a\) and \(b\). The area of this infinitesimal rectangle, denoted by \(dA\), is called the differential element of area or simply the element of area: \[dA=y\,dx=f(x)\,dx.\] Now we conceive the required area as the result of adding up these differential elements of area \(dA\) as the rectangle sweeps across the region. This way of addition is symbolically written as

\[\text{total area }A=\int dA=\int_{a}^{b}y\,dx=\int_{a}^{b}f(x)\,dx.\]

Figure 1
Figure 1

In this practical way of thinking, we skipped the laborious details (dividing the region into rectangles of base \(\Delta x\) and height \(y\), forming the sums, taking the limit, and then switching \(\Sigma\) to \(\int\), \(\Delta x\) to \(dx\), and \(\approx\) to \(=\)), and directly developed the formula for the area under a curve. This intuitive approach to the process of integration is extensively used in science and engineering.

  • Note that if \(A(x)\) denotes the area under the graph of \(f\) between \(a\) and \(x\) (Figure 2), then \[A(x)=\int_{a}^{x}f(t)dt.\] The Fundamental Theorem of Calculus asserts that \[\frac{dA}{dx}=f(x)\] or equivalently \[dA=f(x)dx.\] This is exactly what we expressed for the differential element of area.
    Figure 2
    Figure 2

In this chapter, try to understand the method that we use to construct the formulas instead of just memorizing the formulas. If you understand the method, you can construct the formulas from scratch whenever needed.

 

following sections:

9.1 The area between two curves

9.2 Volumes of solids of revolution: the disk and washer methods

9.3 Volumes of solids with known cross sections: the slice method

9.4 Volumes: the shell method

9.5 Arc length

9.6 Work

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