 Finding area of polygonal regions can be accomplished using area formulas for rectangles and triangles.  Finding area bounded by a curve is more challenging.  Consider that the area inside a circle is the same as the area of an inscribed n-gon where n is infinitely large.

 Summation notation simplifies representation  Area under any curve can be found by summing infinitely many rectangles fitting under the curve.

 Riemann sum is the sum of the product of all function values at an arbitrary point in an interval times the length of the interval.  Intervals may be of different lengths, the point of evaluation could be any point in the interval.  To find an area, we must find the sum of infinitely many rectangles, each getting infinitely small.

 Let f be a function that is defined on the closed interval [a,b].  If exists, we say f is integrable on [a,b]. Moreover, called the definite integral (or Riemann integral) of f from a to be, is then given as that limit.

 The definite integral from a to b of f(x) gives the signed area of the region trapped between the curve, f(x), and the x-axis on that interval.  The lower limit of integration is a and the upper limit of integration is b.  If f is bounded on [a,b] and continuous except at a finite number of points, then f is integrable on [a,b]. In particular, if f is continuous on the whole interval [a,b], it is integrable on [a,b].

 Polynomial functions  Sin & cosine functions  Rational functions, provided that [a,b] contains no points where the denominator is 0.

 Let f be continous on the closed interval [a,b] and let x be a (variable) point in (a,b). Then

 The rate at which the area under the curve of function, f(t), is changing at a point is equal to the value of the function at that point.

 Let f be continuous (integrable) on [a,b], and let F be any antiderivative of f on [a,b]. Then the definite integral is

 Let g be a differentiable function and suppose that F is an antiderivative of f. Then

 It is the chain rule! (from differentiation)  In this case, you have an integral with a function and it’s derivative both present in the integrand.  This is often referred to as “u-substitution”  Let u=function and du=that function’s derivative

 Let g have a continuous derivative on [a,b], and let f be continuous on the range of g. Then where u=g(x ):

 For a definite integral, when a substitution for u is made, the upper and lower limits of integration must change. They were stated in terms of x, they must be changed to be the corresponding values, in terms of u.  When this change in the upper & lower limits is made, there is no need to change the function back to be in terms of x. It is evaluated in terms of the upper & lower limits in terms of u.

 Average Value of a Function: If f is integrable on the interval [a,b], then the average value of f on [a,b] is:

 If you consider the definite integral from over [a,b] to be the area between the curve f(x) and the x-axis, f-average is the height of the rectangle that would be formed over that same interval containing precisely the same area.

 If f is continuous on [a,b], then there is a number c between a and b such that

 If f is an even function then  If f is an odd function, then

 If f is continuous on a closed interval [a,b], then the definite integral must exist. However, it is not always easy or possible to find the definite integral.  In these cases, we use other methods to closely approximate the definite integral.

 Left (or right or midpoint) Riemann sums (estimate the area with rectangles)  Trapezoidal Rule (estimate with several trapezoids)  Simpson’s Rule (estimate the area with the region contained under several parabolas)

 Approximating the definite integral of f(x) over the interval from a to b.