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1 Defiite Itegrls & Numeric Itegrtio Show ll work. Clcultor permitted o, 6,, d Multiple Choice. (Clcultor Permitted) If the midpoits of equl-width rectgles is used to pproximte the re eclosed etwee the x-xis d the grph of y= x x, the pproximtio is (A) (B). (C).666 (D).7 (E). If f ( x) dx = 8, the ( ) ( ) f x + dx = (A) (B) (C) (D) (E) x dx=. ( ) (A) (B) (C) 8 (D) 6 (E). If f (x) dx = +, the ( f x +) ( ) dx = (A) + + (B) (C) (D) (E). The expressio K + is Riem sum pproximtio for x x (A) dx (B) xdx (C) dx (D) xdx (E) xdx x 6 f ( x) The tle ove gives the vlues of fuctio otied from experimet. Use them to estimte 6 f x)dx usig three equl suitervls usig ( () right edpoits (REP) () left edpoits (LEP) (c) midpoits (MDPT) (d) the trpezoidl rule (TRAP)
2 Short Aswer (e) If the fuctio is sid to e decresig fuctio, c you sy whether your estimtes re less th or greter th the exct vlue of the itegrl Could y of these estimtes pproximte the re of the eclosed regio with the x-xis Why or why ot 7. Approximte the re of the regio ouded y the grph of y = si x d the x-xis from x = to x = π usig equl suitervls usig () left edpoits () right edpoits (c) midpoits (d) trpezoidl rule 8. The grph of f is show elow. Evlute ech itegrl y iterpretig it i terms of res. ( )dx () f x ( )dx () f x 7 ( )dx (c) f x ( )dx (d) f x. Fid f ( x) dx if ( f x), x < =. (Hit: Sketch the grph d iterpret the res) x, x
3 . Give tht () xdx = 8, usig your kowledge of trsformtios, wht is tdt () ( x + ) dx (c) x dx (d) xdx. If f ( x ) is represeted y the tle elow, pproximte f ( ).6 x dx usig left-edpoit, right-edpoit, midpoit, d trpezoidl pproximtios. Lel ech oe. Use s my suitervls s the dt llows. x f ( x ) 6 7. Write s sigle itegrl i the form f (x) dx : f (x)dx + f (x)dx f (x)dx. If f ( x). If f (x) dx = d dx = 7 d f (x)dx =.6, fid f (x)dx g (x)dx =6, fid f x) + g(x) (. (Clcultor Permitted) Use your clcultor s fit( fuctio to evlute the followig itegrls. Report decimils. π / x () dx () + x + txdx
4 Σ GENERAL FORM: [ lim f + (i-) x ] x = f ( x) dx. Represet the summtio lim + i s defiite itegrl. f (x) the vlue of the vlue of. Represet the summtio lim s defiite itegrl. + i f (x) the vlue of the vlue of. Represet the summtio lim + + i s defiite itegrl. f (x) the vlue of the vlue of
5 . If is positive iteger, the lim ( ) ( ) ( ) + + c e expressed s wht defiite itegrl Your gol is to write the expded summtio i sigm form, the covert it to defiite itegrl. f (x) the vlue of the vlue of. If is positive iteger, the f (x) lim c e expressed s wht defiite itegrl Your gol is to write the expded summtio i sigm form, the covert it to defiite itegrl. the vlue of the vlue of For questios #6, use the ledig questios ove to help you form defiite itegrl to represet ech summtio. 6. If is positive iteger, the lim c e expressed s wht defiite itegrl 7. If is positive iteger, the π π π si + si si lim c e expressed s wht defiite itegrl 8. If is positive iteger, the wht defiite itegrl lim c e expressed s. If is positive iteger, the lim c e expressed s wht defiite itegrl 6
6 . If is positive iteger, the defiite itegrl lim c e expressed s wht 6. If is positive iteger, the defiite itegrl lim c e expressed s wht. If is positive iteger, the itegrl lim c e expressed s wht defiite. If is positive iteger, the defiite itegrl π π ( ) π cos + cos + cos... + cos lim c e expressed s wht. A. If is positive iteger, the itegrl lim ( ) ( ) ( ) + + c e expressed s wht defiite B. Whe Miss Brow creted this prolem, she strted with the fuctio ( x) = x + how your defiite itegrl swer from prt A is the sme s Miss Brow s defiite itegrl f over the domi [,] x + dx.. Show ( ). A. If is positive iteger, the defiite itegrl lim c e expressed s wht B. Whe Miss Brow creted this prolem, she strted with the fuctio ( x) = x + f over the domi [,] Show how your defiite itegrl swer from prt A is the sme s Miss Brow s defiite itegrl x + dx..
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