Translate the following: I am tired today, and I want to go to my brother's home and cook a big meal for he and I and then sleep on his couch.
.انا تعبان اليوم، وانا اريد ان اذهب الى بيت اخي واطبخ لنا وجبة كبيرة ثم اريد ان انام على اريكته
What is the volume of the solid created by rotating the region bounded by the function f(x) = x^2, the x axis, and the line x=5 around the y-axis?
A basic 2-dimensional integral will show you the area underneath some curve. We know that the region described in the question will have an area of: (0,5) S[ (x^2) ]dx = [x^3/3] (0,5) = 125/3 However, we are concerned with what the solid would look like if you rotated this region around the Y-axis. It is a disk with one concave side. This volume will be easiest to calculate if we do 2 integrals: First, we will calculate what the volume would be without the concave portion. Then, we subtract off the calculated volume of the concave portion in order to find the correct volume. Many people are tempted to use integrals when calculating any volume, but for the first integral here, we have no need. We want to calculate the volume of a disk with a radius of 5 and a height of f(5) =25. Therefore, V = pi*(r^2)*h, or V = pi*(25)(25). Thus, the volume of the disk without the concave depression will be V = 625*pi. Next, we will calculate the volume of the concave portion. This can be thought of as the region bounded by f(x) = x^2 and the y-axis, from y=0 to the maximum height of y=25 (as we calculated in the last part.) To calculate the area of this region, you would want to essentially cut the region into a bunch of infinitely thin horizontal lines and then add them all together. However, since we want the volume of the concave shape created by rotating the region around the y-axis, we will do a really simple and intuitive trick: instead of simply adding the lengths of the lines together as we would to find the area, we will instead use the lines to calculate the area of a bunch of infinitely thin circles and then add those together to render the volume. Whereas the area equation would look like: (0,25) S[ (x)^1/2 ]dx, Instead we will transform the equation inside of the integral into a circle by using it as the radius, and using A = pi*r^2. So: (0,25) S[ pi*(x) ]dx = [pi*(x^2)/2] (0,25) = pi*625/2 And thus volume of the original solid is: 625*pi - pi*625/2 = pi*625/2
Why do software companies often market and sell different versions of their software with varying degrees of functionality, when the marginal cost of distributing any of those versions is exactly the same?
This is a case of price discrimination, specifically second-degree price discrimination. The problem is that companies are not able to determine what a consumer is willing to pay for. Consumer demand may be limited if they only offer the lowest functional software, and if they only marketed the full-functional software, they might be losing value from people who would be willing to pay more for the full functionality. By creating different levels of functionality, the company is essentially creating multiple markets for one good. Consumers who value the high functionality will pay a premium for it, and consumers who only want the lower functionality software will likely have a lower valuation of it. Thus, by price discriminating, firms are able to maximize their profits without incurring any additional cost.