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3.6 Formula Sheet

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, 2007 Antiderivatives (indefinite integrals) x p dx = x p +1 p + 1 for p = - 1 x - 1 dx = ln | x | e rx dx = e rx r Substitution. Suppose G = g . Letting u = 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 ) dx Definite integrals. The area under the curve f between a and b is given by b a f ( x ) dx = F ( b ) - F ( a ) = F ( x ) | b a where F is ANY antiderivative. If we rotate the part of the curve f over the interval [ a, b ] around the x -axis, the result is a solid of revolution with volume V = b a πf ( x ) 2 dx . The improper integral 0 f ( x ) dx = lim b →∞ f ( x ) dx To find the limit it is useful to know that if p, q, r > 0 then as x → ∞ e - rx , x - q , x q e - rx , (ln x ) - p , (ln x ) p x - q 0 A probability density has f ( x ) 0 and f ( x ) dx = 1. The mean
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