Question

1. Find the area between the curve f(x)=sin^3(x)cos^2(x) and y=0 from 0 ≤ x ≤ π

2. Find the surface area of the function f(x)=x^3/6 + 1/2x from 1≤ x ≤ 2 when rotated about the x-axis.

Answer #1

Consider the curve y = e sin x for π /6 ≤ x ≤ π /3 . Set up the
integrals (without evaluating) that represent
1. The area of the surface generated by revolving the curve
about the x-axis.
2. The area of the surface generated by revolving the curve
about the y-axis.

Consider the parametric equations below.
x = t sin(t), y = t
cos(t), 0 ≤ t ≤ π/3
Set up an integral that represents the area of the surface
obtained by rotating the given curve about the x-axis.
Use your calculator to find the surface area correct to four
decimal places

Identify the surface with parametrization x = 3 cos θ sin φ, y =
3 sin θ sin φ, z = cos φ where 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π. Hint: Find
an equation of the form F(x, y, z) = 0 for this surface by
eliminating θ and φ from the equations above. (b) Calculate a
parametrization for the tangent plane to the surface at (θ, φ) =
(π/3, π/4).

Question(9) Curve ? = t - sin t, y = 1 - cos t, 0 ≤ ? ≤ 2? given.
a) Take the derivatives of x and y according to t and arrange them.
b) Write and edit the integral that gives the surface area of the object formed by rotating the given curve around the x axis.
c) Solve the integral and find the surface area.

The
curve x = sqrt( (2y-y ^ 2) ) with 0 <= y <= 1/2 is rotated on
the y axis. Find the surface area of the solid obtained.

If the infinite curve y=e-x, x>/=0 is rotated
about the x-axis, find the area of the resulting surface.
Please show all work.

1) find the
absolute extrema of function f(x) = 2 sin x + cos 2x on the
interval [0, 2pi]
2)
is f(x) = tanx
concave up or concave down at x = phi / 6

1.Find ff if
f′′(x)=2+cos(x),f(0)=−7,f(π/2)=7.f″(x)=2+cos(x),f(0)=−7,f(π/2)=7.
f(x)=
2.Find f if
f′(x)=2cos(x)+sec2(x),−π/2<x<π/2,f′(x)=2cos(x)+sec2(x),−π/2<x<π/2,
and f(π/3)=2.f(π/3)=2.
f(x)=
3.
Find ff if f′′(t)=2et+3sin(t),f(0)=−8,f(π)=−9.
f(t)=
4.
Find the most general antiderivative of
f(x)=6ex+9sec2(x),f(x)=6ex+9sec2(x), where −π2<x<π2.
f(x)=
5.
Find the antiderivative FF of f(x)=4−3(1+x2)−1f(x)=4−3(1+x2)−1
that satisfies F(1)=8.
f(x)=
6.
Find ff if f′(x)=4/sqrt(1−x2) and f(1/2)=−9.

Find the exact area of the surface obtained by rotating the
curve about the x -axis.
y = sin π x/ 5 , 0 ≤ x ≤ 5

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^...

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