APPLICATIONS OF INTEGRATION
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1 Chpter 3 DACS 1 Lok 004/05 CHAPTER 5 APPLICATIONS OF INTEGRATION 5.1 Geometricl Interprettion-Definite Integrl (pge 36) 5. Are of Region (pge 369) 5..1 Are of Region Under Grph (pge 369) Figure 5.7 shows the region ounded y the curve y f (, the x-xis, nd the lines x nd x. This region is locted ove the x-xis. The re of this region is given y f ( dx 1
2 Chpter 3 DACS 1 Lok 004/05 Figure 5.8 shows the region ounded y the curve y g(, the x-xis, nd the lines x nd x. This region is locted elow the x-xis. The definite integrl f ( dx hs negtive vlue. Since the re is lwys positive quntity, the re of this region is written s g( dx Figure 5.9 shows the region ounded y the curve x u(, the y-xis, nd the lines y c nd y d. This region is locted on the right hnd side of the y-xis. The re of this region is given y d c u( dy Exmple 5.3 (pge 370): Find the re of the region ounded y the curve y x, the x-xis nd the lines x 0 nd x 1. Exmple 5.4 (pge 371):
3 Chpter 3 DACS 1 Lok 004/05 Find the re ounded y the lines y x, x 3, x 4 nd x-xis. Exmple 5.8 (pge 374): Find the re of the region in the first qudrnt ounded y the curve y, y-xis with lines y nd y 4. x 5.. Are of the Region etween Two Curves (pge 374) (pge 375) Given two curves y f ( nd y g(, let g( f ( for [, ], the re of the region tht lies etween these two curves in the intervl [, ] (which is locted ove the x- xis) is f ( dx g( dx f ( g( dx The ove formul is lso vlid for the grph tht is locted elow the x-xis. Definition 5. (Are Between Two Curves) (pge 376) If f ( nd g ( re continuous in the intervl [, ] nd g( f ( for ll x in [, ], then the re of the region 3
4 Chpter 3 DACS 1 Lok 004/05 ounded y the curves y f ( nd y g( etween the lines x nd x is given y A f ( g( dx. Exmple 5.10 (pge 376): Find the re of the region ounded y the curve y x nd the lines y x, x 0nd x 1. Exmple 5.15 (pge 380): Find the re of the region ounded y the curves y 3 x nd the line y x 1. Definition 5.3 (Are Between Two Curves out y-axis) (pge 383) If u ( nd v ( re continuous in the intervl [c, d] nd v( u( for ll y in [c, d], then the re of the region ounded y the curves x u( nd x v( etween the lines y c nd y d is given y d A u( v( dy. c Exmple 5.17 (pge 383): 4
5 Chpter 3 DACS 1 Lok 004/05 Find the re of the region ounded y the curves y 3 x nd line y x Volume of Revolution (pge 387) If plne region is revolves out line then solid oject is generted which is clled the solid of revolution, nd the line is clled the xis of revolution. Definition 5.4 (Volume of Revolution out x-axis) (pge 388) Let f ( e non-negtive nd continuous function in the intervl [, ]. If the region etween this curve, the x-xis nd the lines x nd x revolves 360 out the x-xis, then the volume of the solid generted is V f ( dx. Exmple 5.0 (pge 388): 5
6 Chpter 3 DACS 1 Lok 004/05 Find the volume of the solid of revolution when the region ounded y the prol y x nd x-xis within the intervl [0, 4] revolves 360 out the x-xis. Definition 5.5 (Volume of Revolution out y-axis) (pge 390) Let u ( e non-negtive nd continuous function in the intervl [c, d]. If the region ounded y x u(, the y-xis nd the lines y c nd y d revolves 360 out the y-xis, the volume of the solid generted is V d u( dy. c Exmple 5.3 (pge 390): Find the volume of the solid of revolution when the region ounded y the curve y x 1, the lines y 1, y nd the y-xis revolves 360 out the y-xis. Definition 5.6 (Volume of Revolution out x-axis etween two Curves) (pge 391) Let f ( nd g( e non-negtive nd continuous functions in the intervl [, ] nd g( f ( for ll x in the intervl [, 6
7 Chpter 3 DACS 1 Lok 004/05 ]. The volume of revolution when the region ounded y y f (, g (, x nd x, revolves 360 out the x- xis is V f ( g( dx. Exmple 5.4 (pge 391): Find the volume of the solid of revolution when the region ounded y the curve out the x-xis. y 8x nd y x revolves t 360 Definition 5.7 (Volume of Revolution out y-axis Between Two Curves) (pge 39) Let u ( nd v( e non-negtive nd continuous function in the intervl [c, d] nd v( u( for ll y in the intervl [c, d]. The volume of the solid generted when the region ounded y the curves x u(, x v(, y c nd revolves 360 out the y-xis is d c u( v( V dy. y d Exmple 5.5 (pge 39): Find the volume of the solid of revolution when the region ounded y the curve out the x-xis. y 8x nd y x revolves t 360 7
8 Chpter 3 DACS 1 Lok 004/05 Exercise t home: (Tutoril 10) (Pge 385) Quiz 5B: no. 1, 3, 4, 7. (Pge 394) Quiz 5C: no. 1(), 1(c), (), (e). (Pge 395) Exercise 5: no. 3, 33, 43. 8
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