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Unformatted text preview: 1.  Approximated By Rectangles 2.  Calculated By Area of Shapes 3.  Anti-derivatives Integration Approximated by Rectangles (here 4 rectangles) http://en.wikipedia.org/wiki/Riemann_sum Left Endpoints http://en.wikipedia.org/wiki/Riemann_sum Right Endpoints Add up the area of all the rectangles lim+lim eln(= = limy y= lim+ xx = 11 e y ) lim =x!0 + x = lim x! 0 x ! 0+ + + x! 0 x!0 x!0 100100 ✓ X X✓ ◆ ◆ 11 11 b i i i i +11 + i=1 i=1 a n X n X i, i=1 n X If we use enough rectangles… n X n n i,X i2 , X i3 i2 , i=1 i3 i=1 Area of each Zb n rectangle http://upload.wikimedia.org/wikipedia/commons/2/2a/ X Riemann_sum_convergence.png f (x) dx = lim n f (xi ) x Z i=1 i=1 ba i=1 f (x) dx = lim n!1 a where X i=1 n!1 x= b a n f ( xi ) x i=1 , xi = a + i x Definition of an integral as a Reimann sum Integration using Shapes Integration Using anti-derivatives ¤༊ See table on page 398 of your textbook for the ones your should have memorized. ¤༊ Substitution i...
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