MATH 27 Lecture Guide Unit 3 for AY 2018-19.pdf - MATH 27...

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MATH 27 Lecture Guide UNIT 3 (IMSP,UPLB) 1 MATH 27 LECTURE GUIDE UNIT 3. APPLICATIONS OF THE DEFINITE INTEGRAL The definite integral has numerous applications especially in the fields of physics, engineering, and probability and statistics. In this unit, we discuss some applications of the definite integral. In particular, we deal with problems involving certain geometric quantities such as the area of a region, the volume and surface area of a solid, the length of an arc, and the center of mass of a region and of a solid. Our goals for this unit are as follows. By the end of the unit, you should be able to solve for the area of a plane region; solve for the volume of the solids of revolution using disk and cylindrical shell methods; identify when to use disk or cylindrical shell method; find centroids of plane area and solids of revolution; find the length of arc of a curve; and solve for the area of surfaces of revolution. Note: It is expected that you already know how to evaluate definite integrals. Most examples and exercise problems will only require setting up the definite integral that will solve for the measure of the object required. In case you forgot, this is how definite integrals are evaluated:RECALL:If ?is continuous on [?,?]and ????=𝑭?+𝑪,then ??????=𝑭?𝑭?.
MATH 27 Lecture Guide UNIT 3 (IMSP,UPLB) 2 ILLUSTRATION: 1.Determine the area of the region bounded by ?=?𝐢? ?and ?=𝐜?? ?, ??????. Solution: When ??????, ?=?𝐢? ?lies above ?=𝐜?? ?. Using vertical strips, the area of the described region is given by =??square units. Note:Using horizontal strips to solve the problem would be difficult. (Why?) 𝑨= ????????= ?𝐢? ?𝐜?? ????????= −𝐜?? ??𝐢? ??????= ??+??− −????3.1 Area of a Plane Region Using vertical strips, Using horizontal strips, Area: Area: ?=???=???=???=??MUST REMEMBER!!! To get area of a plane region, integrate area of a strip. Use the length of the strip for the length and ??or ??for the width. Length Width Vertical strip: ????(??????????) ??Horizontal strip: ????(?????????) ???=?𝐢? ??=𝐜?? ?
MATH 27 Lecture Guide UNIT 3 (IMSP,UPLB) 3 ILLUSTRATION: 2.Using horizontal strips, set-up the definite integral that will give the area of the region bounded by ?=??and ?=??, ??Solution: When ???, ?=??(which is equivalent to ?=𝐥?𝐠??) lies on the right of ?=??(which is equivalent to ?=𝐥? ?). Using horizontal strips, the area of the described region is given by TO DO:1.Using vertical strips, set-up the definite integral that will give the area of the region above. (Hint:Split the region into two. Solve for the points of intersection of ?=??and ?=?and of ?=??and ?=?to get the proper bounds of the integral.) 2.Evaluate the integrals in the previous item and in Illustration 2. Compare the results. 𝑨=????????= 𝐥?𝐠??𝐥? ??????=???=???=??=??.
MATH 27 Lecture Guide UNIT 3 (IMSP,UPLB) 4 For more exercises, you can refer to: Ron Larson & Bruce H. Edwards. (2016). Calculus, 10thEdition. Philippines: Cengage Learning Asia Pte. Ltd., pp. 442-445 James Stewart. (2013). Calculus, 7thEdition. Philippines: Cengage Learning Asia Pte.

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