Recent content by agro

If: x = f(t) (continuous and differentiable) y = g(t) (continuous) x is nondecreasing on [a, b] y is nonnegative on [a, b] Then when we trace the points (x,y) from t=a to t=b, we can calculate the area bounded above by the traced curve (below by y = 0, left by x = f(a), and right by x = f(b))...

Homework Statement With a > 0, b > 0, and D the area defined by D: \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 Change the integral expression below: \iint\limits_D (x^2+y^2) dx\,dy by using x = a r cos θ, y = b r sin θ. After that evaluate the integral. The Attempt at a Solution...

Thanks for the reply lanedance. Your explanation from the viewpoint of operator is interesting. Anyway I tried following the substitution as is, and after playing around a bit I can transform it into a separable form... u' = (1-2u)/x du/dx = (1/x)/(1/(1-2u)) \frac{1}{x}dx + \frac{1}{2u-1}du =...

Hello there. I'm studying for graduate school entrance exam (nagoya university), and analysis is part of it. I've learned calculus on my undergraduate course but since I didn't get differential equations, I'm kinda learning it by myself right now. The questions are from past problems which are...

truth value of "for all x in {}" and "there exist x in {}" Suppose T is a true statement. Now, given a nonempty set A, both the statement for all x in A, T and there exist x in A, T are true. However, let E be the empty set. What is the truth value of for all x in E, T and...

I'm about to read "Naive Set Theory" by Paul R. Halmos. Amazon sells one published by Springer (1st edition, 1998) while my library (Universitas Gadjah Mada, Indonesia) has one published by Princeton (1st edition, 1960). Is the content any different? If it is significantly different I'll try...

Well, we can easily make them couldn't we :)? \begin{array}{rcl}\frac{d^3}{dx^3}C &=& \frac{d}{dx}\frac{d}{dx}\frac{d}{dx}C\\ &=& \frac{d}{dx}\frac{d}{dx}0\\ &=& \frac{d}{dx}0\\ &=& 0\end{array} Thus we get our first theorem: \frac{d^3}{dx^3}C &=& 0 etc etc...

Here is the approach on Howard Anton's Calculus... First define ln(x) as \int_1^x \frac{1}{x} dx. Then by the Fundamental Theorem of Calculus (part 2), d/dx(ln(x)) = 1/x. Now define e^x as the inverse function of ln(x). The derivative is then \frac{1}{d/du(\ln u)|_{u=e^x}} = e^x

Let us define a pool. Viewed from top, we see a rectangle of sides 16 ft and 10 ft. When we look from the side (so the 16 ft side is perpendicular to our view), it is a trapezoid with width 16 ft and 4 ft, 16 ft being the height. It is roughly like this: From top: 16 ft...

Consider this problem: given a function f(x), we wish to find a function F(x) such that F'(x) = f(x). The process is called antidifferentiation and F(x) is called an antiderivative of f(x). This "general" problem arises in various applications. Consider this problem: I have a particle...

Aaaah... now THAT was simple & elegant :)

My last equation gave x_1 approximately 0.8105 so k is approximately 0.7246. How did you derive your equation, StatusX?

Wrong Equation Sorry guys I got my equation wrong. Here's the correct one: We use the fact that A_1 = x_1\sin{x_1} - \int_0^{x_1} \sin x dx and A_2 = \int_{x_1}^{\pi-x_1}\sin x dx - (\pi - 2x_1)\sin{x_1} By setting A1 = A2 and rearranging, we get: (\pi - x_1)\sin x_1 =...

I'll describe a problem from William Lowell Putnam Mathematical Competition (54th, problem A1): There is an image of the graph y = sin x where x ranges from 0 to pi. The graph of y = k where 0 < k < 1 is also drawn there. Clearly y = k intersects y = sin x at two places, call them (x1, k)...

I once read a 1997 physics book. At the end of the 'static electricity' chapter, it explains the mechanism of DNA replication and protein synthesis (in which static electrical force plays a critical role). However, it says something like 'this model has not been seen in action. It is consistent...