query: Why is the surface area of a revolution $\int 2\pi y ds$ instead of $\int 2\pi y dx$?
positive_passages.docid: 7241
positive_passages.text: Areas versus volumes of revolution: why does the area require approximation by a cone?
negative_passages.docid: 6892
negative_passages.text: Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?
negative_passages.docid: 24545
negative_passages.text: A doubt regarding change of variables in Double Integrals.
negative_passages.docid: 110352
negative_passages.text: Why is a full turn of the circle 360°? Why not any other number?
negative_passages.docid: 155148
negative_passages.text: Why is $\sin 30^\circ=\frac{1}{2}$
negative_passages.docid: 120451
negative_passages.text: Why Does Snell's Law use Sine Instead of Angle Values?
negative_passages.docid: 31922
negative_passages.text: Integral of $\sin x \cos x$ using two methods differs by a constant?
negative_passages.docid: 156485
negative_passages.text: How to evaluate $\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$?
negative_passages.docid: 67224
negative_passages.text: Why is the area under a curve the integral?
negative_passages.docid: 109288
negative_passages.text: Prove that $x^3+y^3+z^3-3xyz=1$ defines a surface of revolution
negative_passages.docid: 28890
negative_passages.text: Why does the tangent of numbers very close to $\frac{\pi}{2}$ resemble the number of degrees in a radian?
negative_passages.docid: 105423
negative_passages.text: Why doesn't every integral from 0 to $2\pi$ equal zero?
negative_passages.docid: 90246
negative_passages.text: Area of an ellipse. (Calculus)
negative_passages.docid: 36586
negative_passages.text: Evaluating $\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta$
negative_passages.docid: 81533
negative_passages.text: Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates
negative_passages.docid: 36701
negative_passages.text: $\displaystyle\int_{0}^{2\pi}\frac{1}{a+ \cos\theta}\,d\theta$
negative_passages.docid: 46378
negative_passages.text: Trig substitution; why can we ignore the absolute value?