The definite integral is a convenient notation used the represent the left-hand and right-hand approximations discussed in the previous section. f (x)dx means the area of the region bounded by f , the y -axis and the lines x = a and x = b. Writing f (x)dx is equivalent to writing
f(xk)Δx |
on the interval [a, b] , but it is a much more compact way of doing so. Note also the similarity between the two expressions. This should serve as a clear reminder that the definite integral is just the limit of right-hand and left-hand approximations.
Unlike the indefinite integral, which represents a function, the definite integral represents a number, and is simply the signed area under the curve of f . The area is considered "signed" because according to the method of calculating the areas by subdivisions, the regions located below the x -axis will be counted as negative, and the regions above will be counted as positive. Negative regions cancel out positive regions, and the definite integral represents the total balance between the two over the given interval. For example, find
sin(x)dx |
Based on the picture of the region being considered, it should be clear that the answer is zero. Here, the negative region is exactly the same size as the positive region:
The definite integral has certain properties that should be intuitive, given its definition as the signed area under the curve:
f (x)dx = f (x)dx + f (x)dx |
f (x)dx=F(b) - F(a) |
x ^{2} dx = (1)^{3} - (0)^{3} = |
x ^{2} dx = x ^{3} = |
v(t)dt=s(b) - s(a) |