Tutor profile: Brenden M.
How do you find the volume of an object in 3-D space?
In order to properly calculate the volume of an unknown object in 3-D space, we must use a triple integral. The integrals are dependent on f(x,y,z)dv which can be simplified as f(x,y,z)dx dy dz. In this case, we integrate with respect to x first, then y, then z; but this order is arbitrary and can be changed. There are 6 possible orders to integrate from, it depends on which way is easiest but all will arrive at the same answer. The tricky part is not necessarily calculating the integral but finding the proper bounds. Typically if the object is defined by a function, let's say f(z), then z will be the first bounds, and your final bounds are typically constants. Once the bounds are set, you simply integrate each as you would a double integral but with one more step.
In physics, we often calculate the motion of an object In a vacuum. But what happens when we look at the motion with real-world effects, especially in regards to the rotational kinematics of the Earth?
The Earth rotates at a rapid rate, about 460 kilometers per hour at the equator, but it moves much slower at the poles, virtually at a rate of zero. As you travel away from the equator the rate at which the Earth spins will slow, which can be found by multiplying the cosine of the degree of latitude by 1,669.8 km/hr. The varying rates at which different latitudinal lines spin has an interesting effect on something we take for granted: the weather. Due to the Earth’s rotation, there is a pattern of deflection taken by objects that are not firmly connected to the ground as they travel long distances around the Earth; this pattern is called the Coriolis Effect. This effect describes how the rotation of the Earth causes air currents to appear to bend away from the pole towards the equator.
Subject: Differential Equations
Studying mathematics goes far beyond solving a problem on a page. So, how are subjects such as differential equations used to analyze and model real-world situations?
Let’s use an example of a car to help easily visualize the math at work. In terms of engineering, engineers attempt to reduce the wear on mechanical systems. For spring-mass systems, reducing the number of oscillations reduces the amount of wear on a system. When looking at oscillating systems we must look at whether the system is over-damped, critically-damped, or under-damped. A system that is over-damped does not oscillate and has a slow response to reaching equilibrium compared to an underdamped system that oscillates to reach equilibrium. In order for the mechanical system to not experience oscillation, it must be critically damped meaning that b =√4mk. In order to find the range b values, we use the maximum and minimum mass values that the car can withstand. With this value, we can find a second-order DE that can be simplified into a first-order DE. This can then be modeled to analyze wear.
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