# Applications of double integrals: A lamina occupies the part of the disk x^2 + y^2 <= 64 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Question
Applications of integrals
Applications of double integrals:
A lamina occupies the part of the disk $$\displaystyle{x}^{{2}}+{y}^{{2}}\le{64}$$ in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

2021-02-28

### Relevant Questions

Applications using double integrals:
A lamina occupies the part of the disk $$\displaystyle{x}^{{2}}+{y}^{{2}}\le{1}$$ in the first quadrant. Use polar coordinates to find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.
Applications of double integrals:
A lamina occupies the part of the disk $$\displaystyle{x}^{{2}}+{y}^{{2}}\le{4}$$ in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.
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1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.
True or False: In many applications of definite integrals, the integral is used to compute the total amount of a varying quantity.
A wood cube 0.30 m on each side has a density of 700kg/m3 and floats levelly in water. (a) What is the distance from the top of the wood to the water surface? (b) What mass has to be placed on top of the wood so that its top isjust at the water level?
The rotor (flywheel) of a toy gyroscope hasmass 0.140 kg. Its moment of inertia about its axis is . The mass of the frame is 0.0250 kg. The gyroscopeis supported on a single pivot javascript:void(0); with its center of mass a horizontaldistance of 4.00 cm from the pivot. The gyroscope is precessing ina horizontal plane at the rate of one revolution in 2.20 s. Find the upward force exerted by thepivot. Find the angular speed with which the rotoris spinning about its axis, expressed in rev/min. Comments
Let C be a circle, and let P be a point not on the circle. Prove that the maximum and minimum distances from P to a point X on C occur when the line X P goes through the center of C. [Hint: Choose coordinate systems so that C is defined by
$$x2 + y2 = r2$$ and P is a point (a,0)
on the x-axis with a $$\neq \pm r,$$ use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]
Two basketball players are essentially equal in all respects. In particular, by jumping they can raise their centers of mass the same vertical distance, H. The first player,Arabella, wishes to shoot over the second player, Boris, and forthis she needs to be as high above Boris as possible. Arabella Jumps at time t=0, and Boris jumps later, at time $$\displaystyle{t}_{{R}}$$(his reaction time). Assume that Arabella has not yet reached her maximum height when Boris jumps.
Part A.) Find the vertical displacement $$\displaystyle{D}{\left({t}\right)}={h}_{{A}}{\left({t}\right)}-{h}_{{B}}{\left({t}\right)}$$, as a function of time for the interval $$\displaystyle{0}{<}{t}{<}{t}_{{R}}$$, where $$\displaystyle{h}_{{A}}{\left({t}\right)}$$ is the height of the raised hands of Arabella, while $$\displaystyle{h}_{{B}}{\left({t}\right)}$$ is the height of the raised hands of Boris. (Express thevertical displacement in terms of H,g,and t.)
Part B.) Find the vertical displacement D(t) between the raised hands of the two players for the time period after Boris has jumped ($$\displaystyle{t}{>}{t}_{{R}}$$) but before Arabella has landed. (Express youranswer in terms of t,$$\displaystyle{t}_{{R}}$$, g,and H)