Calculus Volume 2 · 免费预览
极坐标下的面积与弧长
极坐标下的面积是将二分之一乘以 r 的平方对角度积分，而极坐标下的弧长则是将 r 的平方与 r 导数的平方之和开平方后对角度积分，圆和心形线是其中的经典计算。
1. The polar area formula
The polar area formula
2. Area between polar curves
Area between polar curves
3. Polar arc length
Polar arc length
4. Numerical polar lengths
Numerical polar lengths
5. Polar area formula
The area swept by r = f of theta from alpha to beta equals which integral?
6. Circle from polar area
Applying the polar area formula to the constant curve r = a over [0, 2 pi] gives which area?
7. Area between curves integrand
The area between polar curves r_outer and r_inner over [alpha, beta] uses which integrand?
8. Polar arc length integrand
The polar arc length integrand is which expression?
9. Intersections for area ranges
To compute the area inside the cardioid r = 1 + cos theta but outside the circle r = 1, what must be found first?
10. Rose petal area
One petal of r = cos 2 theta has area pi over 8. As a fraction of pi, what is it? Answer like pi/8.
11. Cardioid perimeter
The cardioid r = 1 + cos theta has perimeter 8. As a number, what is it? Answer with a number.
12. Circle derivative term vanishes
For the circle r = a constant, the arc length integrand simplifies because dr/d theta vanishes. What does the integrand become?
13. Area element sector
The infinitesimal polar area element is half r squared d theta. As a two-word name, what shape is it? Answer like small sector.
14. Radial motion contribution
In the arc length integrand, the term (dr/d theta)^2 represents the contribution of which motion? Answer with two words.