Question 3 Consider the region under the curve y = f ( x ) = 1 / x defined by x

Question

Question 3 Consider the region under the curve y=f(x)=1/x defined by x?1 and 0?y?1/x.

a Sketch the region and show that the area of this region is infinite.

b Write down the improper integral representing the volume when this region is rotated around the x-axis.

c Compute the improper integral found in part b.

d Is the volume finite or infinite?

Considering f a continuous function: If f(x)dx = 1, find (2x)dx. If f(x)dx = 4, find xf(x-)dx. If f(1) = The velocity of a rocket at time t is given by the equation where g is the constant accelaration due to gravity, m is the constant initial onboard fuel mass, r is the constant rate of fuel consumption, k is the constant exhaust speed of What is the velocity of the rocket at time t = ? Give the range off where the equation for v(t) is valid, assuming Find the height, h, of the rocket as a function off (consider h() = ). Consider the region under the curve y = f(x) = 1/z defined by and Sketch the region and show that the area of this region is infinite. Write down the improper integral representing the volume when this region is rotated around the x-axis. Compute the improper integral found in part b. Is the volume finite or infinite?

Explanation / Answer

Question1

a) 5

b) 2

c) -2

question2

a) 0

b) t < m/r

c) -gt^2/2 - k[(m/r - t)*ln((m-rt)/m) + t]

Question3

b)integral [ pi y^2 dx]

= pi*1/x^2 dx

= -pi/x

c) -pi/x|x=1 tox

= pi*(1-1/x)

d) the volume is finite

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