Section 6.4

Let f(t) be a continuous function on the closed interval [a, b]. If I divide the interval into n equal subintervals, each piece will be length For each subinterval, pick a point. (If I were using the right-hand method, I would pick the right endpoint of the interval.) So out of interval I, pick the point x_i. At that point, I will go up to the function and build a rectangle whose height is f(x_i). The area of that rectangle is . We can add all the areas of all of the rectangles: . So if we let the number of rectangles increase without bound we have

This is called the definite integral of f(x) with limits of integration a and b.

Example:

This is the same as asking the area between f(t) = t + 3 and the t-axis on the interval [1, 4]. The graph of this is this. We are looking for the area of the trapezoid. This is a rectangle (base = 3, height = 4) topped by a triangle (base = 3, height = 3).

Area of the trapezoid = area rectangle + area of triangle = 12 + 9/2 = 33/2

Example: Evaluate

Plot y = 3t + 2

We are looking for the area of the gray region.

rectangle (base = 5, height = 2), triangle (base = 5, height = 17- 2)

Example: Evaluate

Plot

This is asking for the area of one-fourth of a circle with radius 2

Example:

When the function is negative, the “area” is negative. So we would have a negative area plus a positive area. The 2 triangles have the same magnitude of area.