(b) Find the volume of the solid generated when R is revolved about the x-axis.
In order to find the volume of a region between y = f (x) and y = g(x), from x = a to x = b,
when it is revolved about the x-axis, we use the following formula:
Here our integral is .
Evaluating the integral, we get
(c) Find the volume of the solid generated when R is revolved about the line x = −1.
In order to find the volume of this region, if we want to use vertical slices, we will use the
method of cylindrical shells. Also, because we are revolving about the line x = −1, we will
need to add 1 to the radius of the cylindrical shell. We will use the formula
2π (x + 1)[f(x) − g(x)]dx
We get
2π (x + 1)[(4 − x^2 )−ex] dx
We suggest that you use your calculator to evaluate the integral.
2π (x + 1)[(4 − x^2 ) − ex] dx = 2π [4x − x^3 − xex + 4 − x^2 − ex]dx = 17.059
- A body is coasting to a stop and the only force acting on it is a resistance proportional to its
speed, according to the equation ; s(0) = 0, where v 0 is the body’s
initial velocity (in m/s), vf is its final velocity, m is its mass, k is a constant, and t is time.
(a) If a body, with mass m = 50 kg and k = 1.5 kg/sec, initially has a velocity of 30 m/s, how
long, to the nearest second, will it take to slow to 1 m/s?
We simply plug into the formula and solve for t.
