3.6 Formula Sheet - Formula sheet for Prelim 3 in Math 106...

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Formula sheet for Prelim 3 in Math 106 on Tuesday April 24, 2007Antiderivatives(indefinite integrals)xpdx=xp+1p+ 1forp=-1x-1dx= ln|x|erxdx=erxrSubstitution.SupposeG=g. Lettingu=f(x),du=f(x)dxg(f(x))f(x)dx=g(u)du=G(u) =G(f(x))Integration by partsf(x)g(x)dx=f(x)g(x)-f(x)g(x)dxDefinite integrals.The area under the curvefbetweenaandbis given bybaf(x)dx=F(b)-F(a) =F(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=baπf(x)2dx.The improper integral0f(x)dx= limb→∞f(x)dxTo find the limit it is useful to know that ifp, q, r >0 then asx→ ∞e-rx,x-q,xqe-rx,(lnx)-p,(lnx)px-q0A probability density hasf(x)0 andf(x)dx= 1. The meanμ
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