INTEGRALS Mth Contents. Anti-derivtives nd indefinite integrls 2. Riemnn sums, definite integrls 2.. Min properties: 2 3. The fundmentl theorem of integrl clculus 2 3.. Corollry (differentition of n integrl with vrible its) 2 3.2. Corollry (the Newton-Leibniz formul; clcultion of definite integrls) 2 4. Applictions of definite integrls 2 5. Geometric pplictions 2 5.. Are of plne region 2 5.2. Arc length 3 5.3. Volume of solid of revolution 3 5.4. Are of surfce of revolution 3 5.5. Generl volume formul 4 6. Applictions to physics [dvnced] 4 6.. Work 4 6.2. Fluid force 4 6.3. Moments, mss, center of grvity 4 7. Techniques of integrtion 4 8. Improper integrls [dvnced] 4 8.. Exmple 4 8.2. Exmple 5 8.3. Independence of the its 5 8.4. The domintion test 5 8.5. The it comprison test 5. Anti-derivtives nd indefinite integrls A function F (x) is clled n nti-derivtive of f(x) if F (x) = f(x). Any continuous function hs ntiderivtives. Any two nti-derivtives of given function differ by constnt. The set of ll nti-derivtives of f(x) is clled the indefinite integrl of f(x); the typicl nottion is f(x) = F (x) + C, where F (x) is one of the nti-derivtives. Importnt Remrk: As in the cse of derivtives, the vrible of integrtion (indicted by the pttern) is very importnt! In n expression like (x + x 2 )d the letter x should be treted s constnt; thus, (x + x 2 )d = 2 2 x + x 2 + C, wheres (x + x 2 ) = 2 x2 + 3 x3 + C. 2. Riemnn sums, definite integrls The expression n i= f(c i) x i is clled the Riemnn sum of f(x). (Here = x < x < x 2 <... < x n = b is prtition of segment [, b], c i [x i, x i ] re some points, nd x i = x i x i.) The it mx x i i= n f(c i ) x i is clled the definite integrl of f(x) from to b nd is denoted by f(x). The vrible of integrtion (= dummy vrible) is importnt (see the remrk in ), lthough it disppers in the result: the result of evlution of definite integrl in respect to x must not contin x! For exmple, (x + x 2 )d = 2 x + x2 (no ), (x + x 2 ) = 2 + 3 (no x).
2 MATH If f is continuous on [, b], the integrl exists. (See lso 8 for improper integrls.) Geometriclly, the integrl is the (signed) re of the region bounded by the grph of f, the x-xis, nd the verticl lines x = nd x = b. 2.. Min properties:. () f(x) = (definition); (2) f(x) = f(x) (definition); b (3) f(x) + c b f(x) = c f(x) ; (4) c f (x) + c 2 f 2 (x)) = c f (x) + c 2 f 2(x), where c, c 2 = const; (5) if b nd f(x) g(x) for ll x [, b], then f(x) g(x) ; (6) if b nd m f(x) M for ll x [, b], then m(b ) f(x) M(b ); (7) if f is continuous on [, b], then there is point c [, b] such tht f(x) = f(c)(b ) (the men vlue theorem). If f(x) is continuous, then 3. The fundmentl theorem of integrl clculus d x f(t) dt = f(x), i.e., the integrl with vrible upper it is n nti-derivtive of f. 3.. Corollry (differentition of n integrl with vrible its). d ψ(x) ϕ(x) f(t) dt = f(ψ(x))ψ (x) f(ϕ(x))ϕ (x). This formul is direct consequence of the fundmentl theorem, chin rule, nd property 2(2). y dt dy Exmple. Given tht x = (t 2, find. We use the formul bove nd implicit differentition to + ) get = y /(y 2 + ); hence, y = y 2 +. 3.2. Corollry (the Newton-Leibniz formul; clcultion of definite integrls). f(x) = F (b) F () = F (x), where F is ny nti-derivtive of f. This is our principl tool for evluting definite integrls. b Importnt Remrk: In pplictions one often needs to find n integrl of the form f(x), wheres it is not esy to find n nti-derivtive of f(x) (i.e., function involving bsolute vlue). In this cse one subdivides [, b] into smller intervls [ i, b i ] (by the roots of f) so tht f keeps sign within ech [ i, b i ] nd finds the integrl s the sum of i i ±f(x). 4. Applictions of definite integrls Assume tht we wnt to clculte quntity S given by the nïve lw Ax (which holds whenever A does not depend on x). If A does depend on x, A = A(x), then we proceed s follows: divide the intervl [, b] where x chnges into smll subintervls. Within ech smll subintervl x i one cn ssume tht A does not chnge much nd, hence, S i = A(x i ) x i. Pssing to the it, one gets A(x). More precisely, for the integrl formul to hold the error in the pproximte formul S i A(x i ) x i must be much smller thn x i (i.e., of order ( x i ) 2 or higher). Below re some prticulr formuls. 5. Geometric pplictions 5.. Are of plne region. Subdivide the region to represent it s the union/difference of simple regions, ech bounded by the grphs of two functions f(x) g(x) nd two verticl lines x = nd x = b. The re of one such simple region is (f(x) g(x)). Sometimes it is esier to use regions bounded by two curves x = f(y) nd x = g(y), f(y) g(y), nd two horizontl lines y = nd y = b. Then the re is (f(y) g(y))dy (see the remrk in ). Both pproches cn be combined with ech other nd with formuls from elementry geometry (when some of the regions re tringles or rectngulrs).
INTEGRALS 3 Importnt Remrk: Drw picture to visulize the region! The condition f(x) g(x) in the integrl formuls is importnt: otherwise some prts of the region will contribute to the re with the minus sign. In fct, the formul is f(x) g(x) (see remrk in 3). 5.2. Arc length. The length of curve is given by L = ds, where the rc length element ds is given by ds = 2 + dy 2 (infinitesiml Pythgoren theorem ). Here re specil cses: The grph y = f(x), x b : L = The grph x = g(y), y b : L = Prmetric representtion x = x(t), y = y(t), α t β : L = β α + (f ) 2. + (g ) 2 dy. (x ) 2 + (y ) 2 dt. 5.3. Volume of solid of revolution. The volume of the solid generted by revolving the region bounded by the grph of function f(x), the x-xis, nd the verticl lines x = nd x = b is given by: πf 2 (x) (disk method; rottion bout the x-xis), or 2πxf(x) (shell method; rottion bout the y-xis; must hve f(x) ). In the former cse (disk method), the region is sliced into verticl segments (perpendiculr to the xis of revolution); the method pplies to revolution bout ny horizontl xes. The region must be mde of verticl segments with one end on the xis of revolution, nd one hs πr 2, where R = R(x) is the length of the verticl segment through x. In the ltter cse (shell method), the region is lso sliced into verticl segments (which re now prllel to the xis of revolution); the method pplies to revolution bout ny verticl xes. The formul tkes the form 2πd(x)l(x), where l(x) is the length of the verticl segment through x nd d(x) is its distnce from the xis of revolution. Mke sure tht l(x) nd d(x) re positive. Importnt Remrk: Ech method hs its own dvntges nd disdvntges. In the disk method, the sign of R(x) is not importnt, but the region should be djcent to the xis of revolution. If there re holes, the volume is found s difference/sum of volumes of simpler solids. In the shell method, the region is rbitrry, but one should mke sure tht the expressions for l(x) nd d(x) re positive (see remrk in 3). Importnt Remrk: In both cses, void overlps, which my result from region intersecting the xis of revolution. A more complicted region should be subdivided into simple ones s when clculting res. Of course, the disk method cn be pplied to verticl xis nd the shell method cn be pplied to horizontl xis; in this cse, one slices the region into horizontl segments nd integrtes with respect to y. 5.4. Are of surfce of revolution. The re of the surfce generted by revolving curve y = f(x), x b, is given by 2πf(x) + (f ) 2 2πx + (f ) 2 The generl formul (for rbitrry curve) is (rottion bout the x-xis), or (rottion bout the y-xis). where r is the distnce from point of the curve to the xis of revolution nd ds is the rc length element t this point (see rc length). r ds,
4 MATH 5.5. Generl volume formul. The volume of solid is given by S(x), where S(x) is the re of the cross-section through x perpendiculr to the x-xis nd nd b re the x-coordintes of, respectively, the leftmost nd the rightmost points of the solid. 6. Applictions to physics [dvnced] 6.. Work. The work of force F (x) directed long the x-xis on the segment x b is W = F (x). 6.2. Fluid force. The totl force of fluid of density w ginst one side of submerged verticl plte running from depth y = to depth y = b is F = wyl(y) dy, where L(y) is the horizontl extent of the plte (mesured long the plte) t depth y. (In this formul the y-xis is ssumed to point downwrds.) 6.3. Moments, mss, center of grvity. The coordintes (x c, y c ) of the center of grvity of system re given by x c = M x /M, y c = M y /M, where M is the mss of the system nd M x, M y re its moments bout the y-xis nd x-xis, respectively. Here re some specil cses: Thin rod long the x-xis of liner density δ(x). M = δ(x), M x = xδ(x), M y =. Thin wire long the grph y = f(x), x b, of liner density δ(x). M = δ(x) + (f ) 2, M x = xδ(x) + (f ) 2, M y = f(x)δ(x) + (f ) 2. More generlly, the formuls re M = δ ds, M x = xδ ds, M y = yδ ds, see rc length. Thin flt plte bounded by the grphs of f(x) g(x) nd the verticl lines x =, x = b, of surfce density δ(x, y). M = ( f(x) g(x) δ(x, y) dy), M x = ( f(x) g(x) xδ(x, y) dy), M y = ( f(x) g(x) yδ(x, y) dy). Thin flt plte bounded by the curves x = f(y), x = g(y), f(y) g(y), nd the horizontl lines y =, y = b, of surfce density δ(x, y). M = ( f(y) g(y) δ(x, y) ) dy, M x = ( f(y) g(y) xδ(x, y) ) dy, M y = ( f(y) g(y) yδ(x, y) ) dy. In the lst two cses be very creful bout the vrible in respect to which you integrte! (See remrk in.) Sy, in the lst cse, fter the inner integrl (with respect to x, with y treted s constnt) is evluted, the expression must no longer contin x, nd the outer integrtion is in respect to y. More complicted regions should be subdivided into simple ones of one of the two bove forms. Mss nd moments re dditive. Keep in mind tht mss must lwys be positive, while moments cn tke negtive vlues s well. Importnt Remrk: Use symmetry whenever possible! If both the plte nd the density re symmetric with respect to n xis, the corresponding moment is. See integrtion.pdf. 7. Techniques of integrtion 8. Improper integrls [dvnced] A definite integrl is clled improper if either it hs infinite its or the integrnd is discontinuous (or both). To evlute n improper integrl, split the intervl of integrtion into subintervls so tht ech subintervl hs t most one singulrity (i.e., n infinite endpoint or point of discontinuity of the integrnd, which must coincide with one of the endpoints of the subintervl). 8.. Exmple. should be split into x x + (, ) nd (, ) re chosen rbitrrily. x + x +. The subdivision points x In fct, these re so clled double integrls; they re considered in Mth 2
Now n integrl with one singulrity is defined s follows: INTEGRALS 5 f(x) = b f(x) = f(x) = f(x) = b b + f(x) f(x) f(x) f(x) (infinite upper it), (infinite lower it), (f(x) is discontinuous t b), (f(x) is discontinuous t ). If the it exists, the integrl is sid to converge; otherwise it is sid to diverge. An integrl with severl singulrities converges if nd only if so does ech of the integrls over the subintervls. 8.2. Exmple. x 2 = b ( b x = ) = converges; b b x 2 = + x = diverges. Importnt Remrk: Do not try to pply the Newton-Leibniz formul f(x) = F (x) b to discontinuous functions! First, it doesn t mke sense, second, it my give wrong result. For exmple, /x = /x + /x diverges (s x + ln x = ), while the Newton-Leibniz formul would give. In mny cses it is only importnt to know whether the integrl converges or not. The following remrks/tests my help to decide this. (For simplicity we consider singulrity t infinity; the it below cn be replced with point of discontinuity of f.) 8.3. Independence of the its. The integrls f(x) nd f(x) converge or diverge simultneously provided tht f is continuous on [, b]. (Of course, the vlue of the integrl does depend on the its.) 8.4. The domintion test. Assume tht f(x) g(x) for ll sufficiently lrge x. Then () if g(x) converges, so does f(x), nd (2) if f(x) diverges, so does g(x). 8.5. The it comprison test. Assume tht f(x) nd g(x) re positive functions nd f(x) = L, < L <. x + g(x) Then g(x) nd f(x) converge or diverge simultneously. The following integrls re useful for comprison: x p converges for p > nd diverges for p, x p converges for p < nd diverges for p, e x converges for < nd diverges for. [dvnced] This topic hs been omitted or moved to Mth 2