PPT-Section 4.3 – Riemann Sums and Definite Integrals

Riemann Sums a b The rectangles need not have equal width and the height may be any value of f x within the subinterval 1 Partition divide ab into N subintervals. Sigma Notation. What does the following notation mean?. means. the sum of the numbers from the lower number to the top number.. Area under curves. In 5.1, we found that we can approximate areas using rectangles.. SUMS MEMBERS. ANDREA BUTTLE. Worked for SUMS since 2001. Have worked for 37 universities in that time from Solent to Cambridge. Reviewed timetabling at 19 universities. Wrote the SUMS good practice guide to teaching space management 2004. Riemann Sums. -Left, Right, Midpoint, Trapezoid. Summations. Definite Integration. We want to think about the region contained by a function, the x-axis, and two vertical lines x=a and x=b. . a. Section 5.2a. First, we need a reminder of . sigma notation:. How do . we evaluate. :. …and what happens if an “infinity” symbol appears. above the sigma???.  The terms go on indefinitely!!!. FACULTY OF EDUCATION. Mathematics Education Department. Integratıon, fınıte sum and defınıte ıntegral. 1. Orhan TUĞ (PhDc). A. Figure 5.1.8. Figure 5.1.9. Error analysis. Error analysis. Upper and lower estimates of the area. Ellen, . M. egan, Dan. Riemann Hypothesis. The nontrivial Riemann zeta function zeros, that is, the values of s other than -2,-4,-6….. . s. uch that . δ. (s)=0 all lie on the critical line . Θ. = R[s] = ½ (with real part ½). GIS Application with Web Service Data Access. Introduction – The Problem. Stormwater . utilities are . unique. Runoff can’t be measured. Must be defensible. Impervious area is usually . the . basis for billing. continuous. functions over . closed. intervals.. Sometimes we can find integrals for functions where the function . is discontinuous or . the limits are infinite. These are called . improper integrals. Area and Estimating with Finite Sums. Section 5.2. Sigma Notation and Limits of Finite Sums. Section 5.3. The Definite Integral. Section 5.4. The Fundamental Theorem of Calculus. As the number of rectangles increased, the approximation of the area under the curve approaches a value.. Copyright .  2010 Pearson Education, Inc.. Section 5.3 – The Definite Integral. Definition. Riemann Sums. The sums you studied in the last section are called . Riemann Sums. When studying . area under a curve. , we consider only intervals over which the function has positive values because area must be positive. Conceptually the idea of . area. is simply. “. the product of two linear dimensions. ” . The notion of Riemann Sum is then an extension of this idea to more general situations. However, in the formula. M. ultiple . S. clerosis. SUMS. study@plymouth.ac.uk. . South West Contacts. :. Dr Jenny . Freeman . . 01752 588835. Esther . Fox . .  . 01752 . 587599. . . East Anglia Contacts :. In this Chapter:. . 1 . Double Integrals over Rectangles. . 2 . Double Integrals over General Regions. . 3 . Double Integrals in Polar Coordinates. . 4 . Applications of Double Integrals. . 5 . Triple Integrals.