Hello. I would appreciate it if someone could check my answers. I'm sorry it is so long.
1.) Let R denote the region between the curves y=x^-1 and y=x^-2 over the interval 1<= x <= 10.
a. Set up an integral for the area of R.
My answer: 1.403
b. Find x-bar, the x coordinate of the centroid of R.
My answer: 4.775
c. Set up and evaluate an integral for the volume of revolution of the solid generated when R is revolved about
i. the x-axis
My answer: 1.781
ii. the y-axis
My answer: infinity
2.) The length of a cable is 50 and the weight is 10. A portion of length 40 was hanging over the edge of a tall building and was pulled to the top. How much work was done?
My answer: 3920
3.) Let C denote the curve y= x(4-x), where 0<= x <= 4. Set up the integral for the following. In this case, do not evaluate the integrals.
a. the length of C
My answer: integral from 0 to 4 of sqrt[ 1+ (4-2x)^2] dx
b. the area of the surface generated when C is revolved about
i. the x-axis
My answer: 2pi *integral from 0 to 4 of x(4-x)*sqrt[1+ (4-2x)^2] dx
ii. the y-axis
My answer: 2pi* integral from 0 to 4 of [sqrt(4-y) -2] *sqrt[1+ (-2*sqrt(4-y))^-2] dy
4.) A tank has the shape of a trapezoidal prism. The top is horizontal and the two ends are vertical. The length is 4. The height is 2. The top is a 3-by-4 rectangle. Viewed from an end, the tank looks like the trapezoid shown in the figure below. Assume the tank contains a liquid to a depth of 1. Take the density of the liquid to be p.
a. Set up, but do not evaluate, an integral for the work required to pump the liquid to the top of the tank.
My answer: W= p*integral from 0 to 2 of 4 dy
b. Set up, but for not evaluate, an integral for the fluid force against one end of the tank.
My answer: F= integral from 0 to 4 of p(24) dx
web site:
img361.imageshack.us/img361/559/calctest3nq.png
My work:
web site:
img382.imageshack.us/img382/2565/calcscan5ae.jpg
Thanks
One note: to check your integrals,
http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=calculus&s2=integrate&s3=advanced#reply
works great.
1a. Correct b. agree c.
c. agree d. Rotating about y...it has a beginning radius and an ending radius in the x domain, so the area of revolution is bounded. It has a bounded volume of revolution. Rethink your limits. You probably will have to break up the integral to three volumes, a bottom, middle cylinder, and a top portion.
2. The integral must be weight * dheight. weight for length dx is 10/50 ...no 9.8 is needed. So work= height *d weight* = INT y 10/50dy= 10/50 y^2/2 limits 0,40
Now think to check this: the weight is .4*10, the average height is 40/2, which checks with the above.
3. a. correct i. yes ii. you have dy limits 0,4...I don't understand that. Secondly, rotating about the y axis, the outer radius is 4, the inner radius is y.
4. I don't understand this. Make a thinkness dy, figuure the area as a function of y. Then the volume is area*dy, and weight is rho*area*dy. Let y=0 be the top of the water.
then you raise the water 1+y
work= weight*height. You can check by figuring the total weight, and the average height is raised.
For number 1d I don't understand what the limits would be.
For 1d. Draw a picture of the two functions, make your area, and rotate about y. If you wish, you can post your drawing on the web again.
What would the integral for number 4 be?
Find the slope and y-intercept of each line...
9. y=x
10.y=-x
11.y=-4
12.y=4
I want get information about Calculus and efficiency in the life
sefw
1.) Let's go through each part step by step.
a. To set up an integral for the area of R, we need to find the area between the curves y=x^-1 and y=x^-2 over the interval 1 <= x <= 10. This can be done by finding the integral of the difference between the two curves over the given interval: ∫(x^-1 - x^-2)dx from 1 to 10.
b. To find the x-coordinate of the centroid of R (x-bar), we need to find the average value of x over the interval 1 <= x <= 10. This can be done by finding the integral of x times the difference between the two curves over the given interval, divided by the area of R: (1/Area) * ∫(x * (x^-1 - x^-2))dx from 1 to 10.
c. To set up and evaluate an integral for the volume of revolution of the solid generated when R is revolved about:
i. the x-axis, we need to find the integral of π * (y(x))^2dx from 1 to 10, where y(x) is the equation of the curve between the two curves. In this case, it is y=x^-1.
ii. the y-axis, we need to find the integral of π * (x(y))^2dy from 1 to infinity, where x(y) is the equation of the curve between the two curves. In this case, it is x=1/y.
2.) To find the work done in pulling a portion of length 40 of a cable with length 50 and weight 10 to the top, we can use the formula for work done: Work = force * distance. In this case, the force is the weight of the cable, which is 10, and the distance is the length pulled, which is 40.
3.) Let's break down each part of this question.
a. To set up the integral for the length of the curve C, we need to find the arc length integral: ∫sqrt(1 + (dy/dx)^2)dx from 0 to 4. In this case, the curve C is defined as y = x(4-x) over the interval 0 <= x <= 4.
b. To set up the integral for the area of the surface generated when C is revolved about:
i. the x-axis, we need to find the integral of 2π * y(x) * sqrt(1 + (dy/dx)^2)dx from 0 to 4, where y(x) is the equation of the curve C.
ii. the y-axis, we need to find the integral of 2π * x(y) * sqrt(1 + (dx/dy)^2)dy from 0 to 4, where x(y) is the equation of the curve C.
4.) To set up the integral for the work required to pump the liquid in a trapezoidal prism-shaped tank to the top, we need to consider the weight of the liquid. The weight is equal to the density (p) multiplied by the volume of the liquid (area * height). In this case, the height of the liquid is 1. So, the integral would be W = p * ∫(area * 1)dy from 0 to 2, where area is the area of the cross-section of the tank as a function of y.
To set up the integral for the fluid force against one end of the tank, we can consider the pressure at the end of the tank. The pressure is equal to the density (p) multiplied by the depth of the liquid at that point. In this case, the depth is y. So, the integral for the fluid force would be F = ∫(p * pressure * area)dx from 0 to 4, where pressure is the pressure at the end of the tank as a function of x and area is the area of the cross-section of the tank.