CK-12-Calculus

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 ...

4.7. Integration by Substitution http://www.ck12.org ∫ √ udu=^23 u^32 +C=^23 ( √ 1 +x^3 )^32 +C. While this method of substituti ...

http://www.ck12.org Chapter 4. Integration Integrating Products of Functions We are not able to state a rule for integrating pro ...

4.7. Integration by Substitution http://www.ck12.org Review Questions Compute the integrals in problems #1–10. 1.∫xlnxdx 2.∫ 13 ...

http://www.ck12.org Chapter 4. Integration 4.8 Numerical Integration Learning Objectives Use the Trapezoidal Rule to solve prob ...

4.8. Numerical Integration http://www.ck12.org The area of a trapezoid isA=h(b^12 +b^2 ), whereb 1 andb 2 are the lengths of the ...

http://www.ck12.org Chapter 4. Integration issue concerns the questions about the accuracy of the approximation. In particular, ...

4.8. Numerical Integration http://www.ck12.org Using parabolas in this way produces the following estimate of the area from Simp ...

http://www.ck12.org Chapter 4. Integration Solution: We need to findnsuch that|Errorsim pson|≤ 0. 001 .We start by noting that∣∣ ...

4.8. Numerical Integration http://www.ck12.org We estimated errors for the Trapezoidal Rule. We used Simpson’s Rule to solve pr ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals CHAPTER 5 Applications of Definite Integrals Chapter Outline 5 ...

5.1. Area Between Two Curves http://www.ck12.org 5.1 Area Between Two Curves Learning Objectives A student will be able to: Com ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals Figure 1b Figure 1c Therefore, as the graphs show, it makes se ...

5.1. Area Between Two Curves http://www.ck12.org A= ∫b a[f(x)−g(x)]dx. Example 1: Find the area of the region enclosed betweeny= ...

http://www.ck12.org Chapter 5. Applications of Definite Integrals A= ∫b a[f(x)−g(x)]dx = ∫ 3 − 2 [(x+^6 )−(x (^2) )]dx. Integrat ...