3.3 Class Notes - April 3 Lecture HWK #8 due April 12/13...

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April 3 Lecture HWK #8 due April 12/13 8.1: 2, 9, 13, 21, 38 11.3: 33, 37 11.1: 6, 12, 23 Fundamental theorem of calculus Given any antiderivative F the area under the curve between a and b is given by the definite integral () b b a a fxd x Fb Fa Fx =−= x 0 0.25 0.5 0.75 1 01234 4 28 4 2 0 0 1 .4998 22 2 x x ee ed x −− == + = Volume A solid of revolution is formed by rotating the area under the curve f(x) for a x b around the x-axis Example: f(x) = xr/h for 0 x h h r cone Rotate the area under the curve f(x) for a x b around the x-axis 2 2 3 2 0 0 ( ) integral of area of slice Example. ( ) / 33 b a h h Vf x d x fx x rh xr r x dx hh π ππ = =⋅=
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Sphere f(x) = (r 2 –x 2 ) 1/2 r -r 32 22 2 0 0 4 2( ) 2 33 r r xr Vr x d x r x π ππ ⎛⎞ =− = ⎜⎟ ⎝⎠ Sphere 3 4 Volume of the sphere 3 The derivative of the volume is r = Sphere 3 3 2 4 Volume of the sphere 3 The derivative of the volume is 4 4 3 The surface area! r dr r dr = = Improper integrals 00 2 0 0 2 0 4 2 0 lim 1 2 1 lim 0 so 2 Recall 0.4998 b xx b b xb b x yx y x ed x x ee x d x x −− →∞ →∞ = == + = ∫∫ Improper integrals, 2: powers 1.1 1.1
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This note was uploaded on 11/14/2007 for the course MATH 1106 taught by Professor Durrett during the Spring '07 term at Cornell University (Engineering School).

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3.3 Class Notes - April 3 Lecture HWK #8 due April 12/13...

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