Create Presentation
Download Presentation

Download Presentation
## Section 5.6: Approximating Sums

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Section 5.6: Approximating Sums**Suppose we want to compute the signed area of a region bounded by the graph of a function from x = a to x = b.**Now, back to the bigger challenge...**Maybe we can approximate the area...**Maybe we can approximate the area...**An underestimate!**Maybe we can approximate the area...**An overestimate!**Maybe we can approximate the area...**Looking better!**Maybe we can approximate the area...**Better yet!**Definition of a Riemann Sum**Let the interval [a, b] be partitioned into n subintervals by any n+1 points a = x0 < x1 < x2 < … < xn-1 < xn = b and let xi = xi – xi-1 denote the width of the ith subinterval. Within each subinterval [xi-1, xi], choose any sampling point ci. The sum Sn = f (c1)x1 + f (c2)x2 + … + f (cn)xn is a Riemann sum with n subdivisions for f on [a, b].**Commonly Used Riemann Sums**• Left-hand • Right-hand • Midpoint**The Definite Integral as a Limit**Let a function f (x) be defined on the interval [a, b]. The integral of f over [a, b], denoted is the number, if one exists, to which all Riemann sums Sn tend as as n tends to infinity and the widths of all subdivisions tend to zero. In symbols:**Trapping the Integral, Part I**• Suppose f is monotone on [a, b]. Then, for any positive integer n, • If f is increasing, • If f is decreasing,**Trapping the Integral, Part II**• For any positive integer n, • If f is concave up on [a, b], • If f is concave down on [a, b],**Simpson’s Rule**For any positive integer n, the quantity is the Simpson’s rule approximation with 2n subdivisions.