Question

2. Rotate the semicircle of radius 2 given by y = √(4 − x^2) about the x-axis to generate a sphere of radius 2, and use this to calculate the surface area of the sphere.

3. Consider the curve given by parametric equations x = 2 sin(t), y = 2 cos(t).

a. Find dy/dx

b. Find the arclength of the curve for 0 ≤ θ ≤ 2π.

4.

a. Sketch one loop of the curve r = sin(2θ) and find the area enclosed.

b. Write down, but **do not evaluate**, an integral
that gives the arc length of this loop

Answer #1

With the parametric equation x=cos(t)+tsin(t), y=sin(t)-tcos(t)
, 0 ≤ t ≤ 2π)
Find the length of the given curve. (10 point)
2) In the circle of r = 6, the area
above the r = 3 cos (θ) line
Write the integral or integrals expressing the area of this
region by drawing. (10 point)

If a circle C with radius 1 rolls along the outside of the
circle x2 + y2 = 49, a fixed point P on C traces out a curve called
an epicycloid, with parametric equations x = 8 cos(t) − cos(8t), y
= 8 sin(t) − sin(8t). Use one of the formulas below to find the
area it encloses. A = C x dy = −C y dx = 1/2 C x dy − y
dx

Evaluate C (y + 6 sin(x)) dx + (z2 + 2 cos(y)) dy + x3 dz where
C is the curve r(t) = sin(t), cos(t), sin(2t) , 0 ≤ t ≤ 2π. (Hint:
Observe that C lies on the surface z = 2xy.) C F · dr =

Using MatLab
2. Given the parametric equations x = t^3 - 3t, y = t^2-3:
(a) Find the points where the tangent line is horizontal or
vertical (indicate which in a text line)
(b) Plot the curve parametrized by these equations to
confirm.
(c) Note that the curve crosses itself at the origin. Find the
equation of both tangent lines.
(d) Find the length of the loop in the graph and the area
enclosed by the loop.
3. Use what...

1. Graph the curve given in parametric form by x = e t sin(t)
and y = e t cos(t) on the interval 0 ≤ t ≤ π2.
2. Find the length of the curve in the previous problem.
3. In the polar curve defined by r = 1 − sin(θ) find the points
where the tangent line is vertical.

Consider the parametric curve given by the equations:
x = tsin(t) and y = t cos(t) for 0 ≤ t ≤ 1
(a) Find the slope of a tangent line to this curve when t =
1.
(b) Find the arclength of this curve

1. This question has several parts that must be completed
sequentially. If you omit a part of the question, you will not
receive any points for the omitted part, and you will not be able
to return to the omitted part. Tutorial exercise Find dy / dx y d 2
y / dx 2, and find the slope and concavity (if possible) at the
given value of the parameter. Parametric equations Point
x = 4 cos θ, y = 4...

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

Given parametric equations below, find d^2y/dx^2 and determine
the intervals on which the graph of the curve is concave up or
concave down.
(a) x = t^2 , y = t^3−3t
(b) x = cos(t), y = sin(2t)

1. Solve the following differential equations.
(a) dy/dt +(1/t)y = cos(t) +(sin(t)/t) , y(2pie) = 1
(b)dy/dx = (2x + xy) / (y^2 + 1)
(c) dy/dx=(2xy^2 +1) / (2x^3y)
(d) dy/dx = y-x-1+(xiy+2) ^(-1)
2. A hollow sphere has a diameter of 8 ft. and is filled half way
with water. A circular hole (with a radius of 0.5 in.) is opened at
the bottom of the sphere. How long will it take for the sphere to
become empty?...

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