# Math 27 lecture guide unit 3 imspuplb 13 illustration

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MATH 27 Lecture Guide UNIT 3 (IMSP,UPLB) 13 ILLUSTRATION 1. Set-up the definite integrals that will give the coordinates of the centroid of the region bounded by ? = ?𝐢? ? and ? = 𝐜?? ? , ? ? ? ?? ? . Solution: We use vertical strips. (Why?) When ? ? ? ?? ? , ? = ?𝐢? ? lies above ? = 𝐜??? . Hence, we have ? = ? [ ? ? ? ? ] ?? ? ? = ? ?𝐢? ? 𝐜?? ? ?? ?? ? ? ? ? ? = ? ? [ ? ? ? ? ] ?? ? ? = ? ? ?𝐢? ? 𝐜?? ? ?? ?? ? ? ? ? ? = ? ? ? ? ? ? [ ? ? ? ? ] ?? ? ? = ? ?𝐢? ? 𝐜?? ? ? ?𝐢? ? 𝐜?? ? ?? ?? ? ? ? . Thus, the centroid of the described region is at ? ̅, ? ̅ = ? ? ? , ? ? ? . Consider the region bounded by = and = , ≤ ≤ . Suppose the area density is . (Simplification: constant area density) Total mass: = Moments of mass: = = Center of mass: ̅, ̅ , where ̅ = and ̅ = Using vertical strips, = [ ] = [ ] = [ ] In general, using horizontal or vertical strip, = = = , where . ? = ? ? ? = ? ?
MATH 27 Lecture Guide UNIT 3 (IMSP,UPLB) 14 ILLUSTRATION 2. Using horizontal strips, set-up the definite integrals that will give the coordinates of the centroid of the region bounded by ? = ? ? and ? = ? ? . Solution: The curves ? = ? ? and ? = ? ? intersect at the points ? , ? and ? , ? , and for the described region, ? ? ? and ? ? ? . Now, note that ? = ? ? is equivalent to ? = ± ? . However, we choose ? = ? since ? ? ? for the described region. On the other hand, ? = ? ? is equivalent to ? = ? ? . Hence, using horizontal strips, we have ? = ? [ ? ? ? ? ] ?? ? ? = ? ? ? ? ?? ? ? ? ? = ? ? ? ? ? ? [ ? ? ? ? ] ?? ? ? = ? ? ? ? ? ? ? ? ?? ? ? ? ? = ? ? [ ? ? ? ? ] ?? ? ? = ? ? ? ? ? ?? ? ? . Thus, the centroid of the described region is at ? ̅, ? ̅ = ? ? ? , ? ? ? . For more exercises, you can refer to: Ron Larson & Bruce H. Edwards. (2016). Calculus, 10 th Edition. Philippines: Cengage Learning Asia Pte. Ltd., pp. 494-496 James Stewart. (2013). Calculus, 7 th Edition. Philippines: Cengage Learning Asia Pte. Ltd., pp. 585-586 For an online tutorial, follow this link: TRY THIS! Set-up the definite integrals that will give the coordinates of the centroid of the following regions. If it is “easy” to evaluate the integrals, determine the coordinates of the centroid of the region. 1. the region bounded by ? = − ? + ? , ? = ? ? , and the ? axis 2. the region bounded by ? ? = ? and ? = − ? + ? 3. the region bounded by ? = ? ? and ? = 𝐥? ? , ? ? ? 4. the circle defined by the equation ? ? + ? ? = ? ? , where ? is a positive constant 5. the semi-circle bounded by ? = ? ? ? ? (where ? is a positive constant) and the ? axis ? = ? ? ? = ? ?
ILLUSTRATION 1. Let ? be the region in the first quadrant bounded by ? = 𝐜?? ? , the ? axis, and the ? axis. Set-up the definite integrals that will give the coordinates of the centroid of the solid generated by revolving ? about the line ? = − ? . ̅ , . . ? = ? and ? ? = 𝐜??? ? = 𝐜?? ? + ? ,
MATH 27 Lecture Guide UNIT 3 (IMSP,UPLB) 15 ? respectively. Noting that for the described solid, ? ? ? ? , we obtain its volume to be