Could anyone please help us out with these problems even just to show how to begin. It would be great and I would be very grateful. I have an exam coming up and I need to get these problems sorted.
Show that the volume of the solid obtained by rotating the part of circle of x^2 + y^2 = 1 that lies between x =0 and x = 1/2 about the x axis is (11pi)/24 cubic units.
Find the antiderivative between e and 1 of (4 ln(x) + 5) / x dx using substitution of w = 4ln(x) + 5
A searchlight rotates at a rate of 3 revolutions per min. Its beam of light hits a wall that is 10km away and produces a dot of light that moves horizontally along the wall. At what speed is the dot moving when the angle between the beam of light and the line through the searchlight perpendicular to the wall is pi/6
Find the slope of the curve x^2 + xy + 2y^3 = 4 at the point (-2,1)
the hint was already given..
you are given with and a constant distance ..
you should make a relationship between and the other variable that i'll call .. in fact, the relationship could be ... do you get the picture?
find and evaluate at that point..
For the first one it might be cooler to parameterize your curve. Let
and
let
Now we know that volume is given by
So seeing that
when
and when
So we would have
Now obviously
So we have our volume is given by
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For your second one I belive you have
If we let as you suggested
and then seeing that it is probably actually easier to not change the limits and just integrate this normall, sub back in and get our integral. So we would just compute
So subbing back in we get
Which implies that
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For the last one we must use implicit differentiation.
So for
Differentiating, implicity mind you, we get
Now we need to find at
So instead of solving for y' which is what you usually do when you are just asked to find the general formula, just plug in the numbers and solve for y' numerically
so we would have
Solving gives us
3 rev per min =A searchlight rotates at a rate of 3 revolutions per min. Its beam of light hits a wall that is 10km away and produces a dot of light that moves horizontally along the wall. At what speed is the dot moving when the angle between the beam of light and the line through the searchlight perpendicular to the wall is pi/6
We want dx/dt given that