Calculus Volume 2 · 免费预览
曲线的弧长与表面积
弧长把曲线各段的微小斜边积分成总长度公式，而将曲线绕轴旋转会扫掠出一个曲面，其面积由对环向带状部分积分得到。
1. The arc length formula
The arc length formula
2. Symmetry and special cases
Symmetry and special cases
3. Surface area of revolution
Surface area of revolution
4. Work and the physical template
Work and the physical template
5. Arc length integrand
For a curve y = f(x) on [a, b], which integrand gives the arc length?
6. Line segment check
The arc length formula applied to the line y = mx on [0, a] gives what length?
7. Surface area band
A piece of curve of length ds at distance r from the axis of rotation sweeps what band area?
8. Sphere surface area
Rotating the semicircle y = sqrt(r^2 - x^2) about the x axis, what surface area integral gives the sphere?
9. Gabriel's horn paradox
Rotating y = 1/x on [1, infinity) about the x axis gives a surface with infinite area but finite volume. What does this demonstrate?
10. Which curves have closed-form lengths
For which curve does the arc length integral on a finite interval evaluate in elementary closed form?
11. Arc length of a simple curve
For y = x on [0, 1], the arc length integral is the integral of sqrt(2) dx. What length results? Answer like sqrt(2).
12. Slant height formula
The cone's surface area from rotation is pi r times the slant height, a single letter usually used for it. What letter? Answer with one lowercase letter.
13. Sphere from rotation
The sphere's surface area integral 2 pi integral of r dx from -r to r evaluates to what multiple of pi r^2? Answer with a number.
14. ds expression
The arc length element ds equals sqrt of 1 plus a squared derivative. As two letters followed by prime notation, what is that derivative? Answer like f'.