CK-12-Calculus

4.3. The Area Problem http://www.ck12.org We then summed the areas of the rectangles as follows: R 1 =^14 ·f ( 1 4 ) = 641 , R 2 ...

http://www.ck12.org Chapter 4. Integration Useful Summation Formulas We can use the notation to indicate useful formulas that we ...

4.3. The Area Problem http://www.ck12.org The following example shows how we can use these to find the area. Example 2: Show tha ...

http://www.ck12.org Chapter 4. Integration Definition Letfbe a continuous function on a closed interval[a,b].LetPbe a partition ...

4.3. The Area Problem http://www.ck12.org Review Questions In problems #1–2 , find the summations. 1.∑^10 i= 1 i( 2 i− 3 ) 2.∑ni ...

http://www.ck12.org Chapter 4. Integration 4.4 Definite Integrals Learning Objectives Use Riemann Sums to approximate areas und ...

4.4. Definite Integrals http://www.ck12.org Solution: If we partition the interval[ 0 , 3 ]inton=6 equal sub-intervals, then eac ...

http://www.ck12.org Chapter 4. Integration ∫ 3 0 x (^3) dx=^81 4. Before we look to try some problems, let’s make a couple of ob ...

4.4. Definite Integrals http://www.ck12.org and Right-hand sums are frequently used in calculations of numerical integrals becau ...

http://www.ck12.org Chapter 4. Integration 4.5 Evaluating Definite Integrals Learning Objectives Use antiderivatives to evaluat ...

4.5. Evaluating Definite Integrals http://www.ck12.org Using the limit definition we found that∫ 03 x^3 dx=^814 .We now can veri ...

http://www.ck12.org Chapter 4. Integration We first need to divide[a,b]intonsub-intervals of length 4 x=b−na. We letx 0 =a,x 1 , ...

4.5. Evaluating Definite Integrals http://www.ck12.org Note thatF(a) =∫aaf(x)dx= 0 .Hence we have f(c) =F(bb−)−a^0 =Fb−(ba), and ...

http://www.ck12.org Chapter 4. Integration 4.6 The Fundamental Theorem of Calculus Learning Objectives Use the Fundamental Theo ...

4.6. The Fundamental Theorem of Calculus http://www.ck12.org We observe that the regions of interest are in the first and third ...

http://www.ck12.org Chapter 4. Integration g(x) =−x^2 − 2 x+ 1. Solution: The graph indicates the area we need to focus on. ∫ 0 ...

4.6. The Fundamental Theorem of Calculus http://www.ck12.org This application of the Fundamental Theorem becomes more important ...

http://www.ck12.org Chapter 4. Integration By division, we have f(u)≤F(xx)−−Fc(c)≤f(v). Whenxis close toc,then bothf(u)andf(v ...

4.6. The Fundamental Theorem of Calculus http://www.ck12.org (Hint: Examine the graph of the function and divide the interval ac ...

http://www.ck12.org Chapter 4. Integration 4.7 Integration by Substitution Learning Objectives Integrate composite functions Us ...