### Surface of Revolution

Y=2*sqrt[x] y=0, x=3 We are using the following formula 2pi intergal of r(x) times the sqrt of 1 + (f'(x))^2.

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Y=2*sqrt[x] y=0, x=3 We are using the following formula 2pi intergal of r(x) times the sqrt of 1 + (f'(x))^2.

Please see the attached file for the fully formatted problems. Questions pertain to Second order Taylor approximations and integrals for two first order differential equations.

Prove that if x>0, then 1+x/2-x^2/8<(1+x)^(1/2)<1+x/2

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Sum of a series

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Show that the volume enclosed by the surface ___ is equal to ___. Please see the attachment.

Please see the attached file for the fully formatted problem. Show that the average distance of a point of a disk with radius a is...

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Please see the attached file for the fully formatted problem. Find the mass and centroid of a plane lamina with the given shape and density delta, the region bounded by y = x2 and x = y2 delta(x,y) = x2 + y2.

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The inverse cosine function has domain [-1,1] and range [0, pi]. Prove that (cos^-1)'(x) = -1 / sqrt(1-x^2)

Find dy/dx implicitly: x^2+6xy-5y+y^3=12

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For which real values alpha does lim {x -> 0+} x^alpha sin(1/x) exist? It is easy to show using the epsilon - delta definition below that this limit exists for all real alpha >= 1. In fact the limit is zero in this case. The case alpha equals zero is also quite simple and the limit does not exist. Consider the two sequence

Find the vectors T, N, and B at the given point. r(t)=<e^t , e^t sint , e^t cost> , (1,0,1) I'm having problems with this one because it requires lots of calc. I and II and i just can't remember how to do some of this. I can take the first derivative and the second but doing the cross product is giving me trouble.

With this problem im seeking a detailed explanation. I already have some of the answers however i dont see how they were obtained. Can you please solve this and show me how each answer was obtained. My numbers are way off from what they should be. Thanks The quarterback of a football team releases a pass at a heigh

1) Find the absolute extreme values of the function f(x,y) = x^2 + xy - x - 2y + 4 on the region D enclosed by y= -x, x=3, y=0 2) Given a circle of radius R. Of all the rectangulars inscribed in the circle, find the rectangular with the largest area. 3) a) Find the differential df of f(x,y)= x(e^y) b) use the differenti