Question

B.) Let R be the region between the curves y = x^3 , y = 0, x = 1, x = 2. Use the method of cylindrical shells to compute the volume of the solid obtained by rotating R about the y-axis.

C.) The curve x(t) = sin (π t) y(t) = t^2 − t has two tangent lines at the point (0, 0). List both of them. Give your answer in the form y = mx + b ?

D.) For the following function, find the Taylor series representation centered at a = 1 and its radius of convergence f(x) = (2 + x)^5

E.) For the following function, find the Maclaurin series. f(x) = tan^(−1) (x/3)

Answer #1

Let R be the region in the first quadrant enclosed by the curves
x = 0, y = k, and y = tan(x)
where k > 0. Express the area of R as a function of k.

Problem (9). Let R be the region enclosed by y = 2x, the x-axis,
and x = 2. Draw the solid and set-up an integral (or a sum of
integrals) that computes the volume of the solid obtained by
rotating R about:
(a) the x-axis using disks/washers
(b) the x-axis using cylindrical shells
(c) the y-axis using disks/washer
(d) the y-axis using cylindrical shells
(e) the line x = 3 using disks/washers
(f) the line y = 4 using cylindrical...

Find a power series representation for the function.
f(x)=x^3/(x-8)^2
f(x)=SIGMA n=0 to infinity
Determine the radius of convergence
Use a Maclaurin series in this table to obtain the Maclaurin
series for the given function
f(x)=xcos(2x)

Find the MacLaurin series for f(x) = cos(5x^3 ) and its radius
of convergence.
Find the degree four Taylor polynomial, T4(x), for g(x) = sin(x)
at a = π/4.

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉.
a) Determine if F ( x , y ) is a conservative vector field and, if
so, find a potential function for it. b) Calculate ∫ C F ⋅ d r
where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin
π...

et S be the solid obtained by rotating the region bounded by the
curves ?=sin(?2) and ?=0 with 0≤?≤root(?) about the ?y-axis. Use
cylindrical shells to find the volume of S.
Volume =

The hyperbolic cosine function, cosh x = (1/2) (e^x + e^-x).
Find the Taylor series representation for cosh x centered at x=0 by
using the well known Taylor series expansion of e^x. What is the
radius of convergence of the Taylor Expansion?

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

Sketch the region bounded by the given curves. y = 3 sin x, y =
ex, x = 0, x = π/2 Find the area of the region.

Let B be the region bounded by the part of the curve y = sin x,
0 ≤ x ≤ π, and the x-axis. Express (do not evaluate) the volume of
the solid obtained by rotating the region B about the y-axis as
definite integrals
a) using the cylindrical shell method
b) using the disk method

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