As usual, we’ll introduce this topic from a geometric point of view. Geometri cally, deﬁnite integrals are used to ﬁnd the area under a curve.Alternately, you can think of them as a “cumulative sum” — we’ll see this viewpoint later.
Figure 1: Area under a curve
Figure 1 illustrates what we mean by “area under acurve”. The area starts at the left endpoint x = a and ends at the right endpoint x = b. The “top” is the graph of f (x) and the “bottom” is the x-axis. The notation we use to describethis in calculus is the deﬁnite integral �
f (x)dx. The diﬀerence between a deﬁnite integral and an indeﬁnite integral (or an tiderivative) is that a deﬁnite integral hasspeciﬁed start and end points.
Deﬁnition of the Deﬁnite Integral
�b The deﬁnite integral a f (x)dx describes the area “under” the graph of f (x) on the interval a < x < b.
aFigure 1: Area under a curve
Abstractly, the way we compute this area is to divide it up into rectangles then take a limit. The three steps in this process are: 1. Divide theregion into “rectangles” 2. Add up areas of rectangles 3. Take the limit as the rectangles become inﬁnitesimally thin Figure 2 shows the area under a curve divided into rectangles. Noticethat since the rectangles aren’t curved they do not exactly overlap the area. Adding up the areas of the rectangles doesn’t give you exactly the area under the curve, but the two areasare pretty close together. The key idea is that as the rectangles get thinner, the diﬀerence between the area covered by the rectangles and the area under the curve will get smaller.In the limit, the area covered by the rectangles will exactly equal the area under the curve.
Figure 2: Area under a curve divided into rectangles