3.3 Class Notes - April 3 Lecture HWK#8 due April 12/13 8.1...

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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 f x dx F b F a F x = = x 0 0.25 0.5 0.75 1 0 1 2 3 4 4 2 8 4 2 0 0 1 .4998 2 2 2 x x e e e dx = = − + = 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 2 2 3 2 2 2 0 0 ( ) integral of area of slice Example. ( ) / 3 3 b a h h V f x dx f x xr h x r r x r h dx h h π π π π = = = = =
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Sphere f(x) = (r 2 – x 2 ) 1/2 r -r 3 2 2 2 2 0 0 4 2 ( ) 2 3 3 r r x r V r x dx r x π π π = = = Sphere 3 4 Volume of the sphere 3 The derivative of the volume is r π = Sphere 3 3 2 4 Volume of the sphere 3 The derivative of the volume is 4 4 3 The surface area! r d r r dr π π π = = Improper integrals 2 2 0 0 2 2 2 0 0 2 0 4 2 0 lim 1 2 2 2 1 lim 0 so 2 Recall 0.4998 b x x b b x b b x y x y x e dx e dx e e e dx e e dx e dx →∞ →∞ = = = − + = = = Improper integrals, 2: powers
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