Calculating the Volume of a Concave Lens Using Multiple Integration

AI Thread Summary
To calculate the volume of a circular concave lens with radius 2 units and two refracting surfaces defined by z=1/2(x^2+y^2+1) and z=-1/2(x^2+y^2), multiple integration is required. The volume can be determined by integrating over the region R in the xy-plane, specifically the area of the circle with radius 2. The height of the lens between the two surfaces is 1 unit, leading to the volume being the area of the circle multiplied by this height. The area of the circle is calculated as π times the radius squared, which is 4π. This approach effectively utilizes multiple integration to find the volume of the concave lens.
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Homework Statement



A circular concave lens of radius 2 units, has two refracting surfaces described by z=1/2(x2+y2+1) and z=-1/2(x2+y2) What is the volume of the glass?


Homework Equations



Over a region R: V=\int\int\int dA


The Attempt at a Solution



I have no idea how to start this. Polar coordinates, bounded regions I don't know anything apart from the fact that I need to use multiple integration.
 
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A line from z= (1/2)(x^2+ y^2) straight up (parallel to the z axis) to z= (1/2)(x^2+ y^2+ 1) has length 1 so the volume is simply the integral of "1" over the region, in the xy plane, the lens covers. And that is just the 1 times the area of that region! What is the area of a circle of radius 2?
 
The second surface is z= -(1/2)(x^2+ y^2+ 1). Its negative. Even so, I'm doing this question for an exam on monday out of practise for multiple integrals. So how would you do this using integration?
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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