Notes 6PI (Propeties of Definite integrals)

Notes 6PI (Propeties of Definite integrals) - MIT...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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PI. PROPERTIES OF INTEGRALS For ease in using the definite integral, it is important to know its properties. Your book lists the following' (on the right, we give a name to the property): (1) f () dx = - abf( f (s) da integrating backwards (2) f (T)dz =0 (3) f (x)dz = f(z) dx + jf(z)dz interval addition (4) + g)d = f() d + 9g(x) d linearity l f(z)dz = c f ()dx linearity f(z) d < jg(z)dl s if f () g() on [a, b estimation Property (5) is useful in estimating definite integrals that cannot be calculated exactly. Example 1. Show that r-dz < 1.3. Solution. We estimate the integrand, and then use (6). We have z 3 < X on [0, 1]; e VI w = (1+ )o = (2%/i- 1) m1.22 < 1.3 . We add two more properties to the above list. (Q) if(x)dxl :5 f(x)j d . absolute value property Property (6) is used to estimate the size of an integral whose integrand is both positive and
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Notes 6PI (Propeties of Definite integrals) - MIT...

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