Formula sheet for Prelim 3 in Math 106 on Tuesday April 24, 2007Antiderivatives(inde±nite integrals)±xpdx=xp+1p+1forp±=-1±x-1dx=ln|x|±erxdx=erSubstitution.SupposeG±=g. Lettingu=f(x),du=f±(x)dx±g(f(x))f±(x)dx=±g(u)du=G(u)=G(f(x))Integration by parts±f(x)g±(x)dx=f(x)g(x)-±f±(x)g(x)dxDefnite integrals.The area under the curvefbetweenaandbis given by±baf(x)dx=F(b)-F(aF(x)|bawhereFis ANY antiderivative.If we rotate the part of the curvefover the interval [a, b] around thex-axis, the result is asolid oF revolutionwith volumeV=²baπf(x)2dx.The improper integral±∞0f(x)dx= limb→∞±f(x)dxTo ±nd the limit it is useful to know that ifp, q, r >0 then asx→∞e-,x-qqe-,(lnx)-p,(lnx)px-q→0A probability density hasf(x)≥0 and²f(x)dx= 1. The meanμ=EX=²xf(x)dx.The second moment
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This note was uploaded on 11/14/2007 for the course MATH 1106 taught by Professor Durrett during the Spring '07 term at Cornell University (Engineering School).