The integral is an area

In differential calculus the starting question was how to find the slope of a tangent to a curve. Later we saw that this "slope" can have many different interpretations (speed, acceleration, rate of change, ...), depending on the context and the problem at hand. The same will be the case for integral calculus. The question here is how to find the area between the xx-axis and a curve, where the curve is given by the graph of a function:

The "area" also has many interpetations, like the physical work, the distance moved, or the total current through a cross section of wire. One little oddity is that what we are actually interested in is not the normal area, but the signed area — that is, regions below the xx-axis count as negative.

In the gray shaded region above, there are two values we can assign the shaded area, depening on how we deal with the part of the region that is below the xx-axis: the normal area (left) is A=3+4+2=9A=3+4+2=9, while the signed area (right) is A=3+4−2=5A=3+4-2=5. Signed areas are interesting because physical constants like work or current can also be negative, translating in negative areas.

Definition 1

Consider a function ff and two values aa and bb on the xx-axis, where a≤ba \leq b. The signed area of the region bounded by the graph of ff, the xx-axis, and the vertical lines x=ax = a and x=bx = b is denoted by

Equation 1
∫abf(x) dx\int_a^b f(x)\, dx

Integral of ff from aa to bb

(read as ":defthe integral of f from a to b). The values aa and bb are called the limits of integration (specifically, the lower and upper limits, respectively).

Exercise 1
  1. Determine if the integral below is positive or negative
    1. ∫−11x2 dx\int_{-1}^1 x^2\, dx
    2. ∫−31x+1 dx\int_{-3}^1 x+1\, dx
  2. Determine the value of the integral below:
    1. ∫−ππsin⁡(x) dx\int_{-\pi}^\pi \sin(x)\, dx
    2. ∫−224x3 dx\int_{-2}^2 4x^3\, dx

Hint: Draw the graph.

Solution

Draw the graphs!

    1. positive
    2. negative
  2. In both cases the signed area is 00, because both subregions are the same, but one is above, the other below the xx-axis. Thus the sum of the two areas equals 00.