Calculus Volume 1 · 免费预览
力矩与质心
力矩是质量与力臂乘积的积分，可使板在一点处平衡；质心则是总力矩为零之处。机械方面的应用至此完成，并以帕普斯定理收尾。
1. Moments of a plate
Moments of a plate
2. Symmetry and composite regions
Symmetry and composite regions
3. Pappus's theorem
Pappus's theorem
4. Variable density plates
Variable density plates
5. Centroid definition
What is the center of mass of a uniform thin plate, in terms of its moments and area?
6. Symmetry shortcut
A region is symmetric about the vertical line x = 2. What does symmetry tell us about its centroid?
7. Semicircle centroid
The semicircular region of radius r has area pi r^2 over 2 and moment M_x equal the integral of one half (r^2 - x^2) dx from -r to r. What is its centroid's y coordinate?
8. Pappus theorem content
Pappus's theorem states that the volume of a solid of revolution equals what?
9. Triangle centroid
The triangle with vertices (0, 0), (b, 0), and (0, h) has area bh/2. Its centroid lies at what point?
10. Rectangle centroid
The rectangle [0, a] by [0, b] has centroid at half each side. What is x bar? Answer like a/2.
11. Composite region centroid
A square plate of side 2 with centroid (1, 1) has a unit square of centroid (1.5, 1.5) removed from its corner. The remaining area is 3. What is the x coordinate of the new centroid?
12. Moment of a strip
A vertical strip of height f(x) and width dx at position x has area f(x) dx. Its moment about the y axis is x times that area. As an expression, what is it? Answer like x f(x) dx.
13. Semicircle moment integral
The integral from -r to r of (r^2 - x^2) dx equals 4r^3 over 3. What is the value of M_x, one half of that? Answer like 2r^3/3.
14. Pappus radius
Pappus's theorem for rotation about the y axis uses the centroid's x coordinate in the travel 2 pi x bar. As a single symbol, what is that coordinate? Answer like x with a bar described, using the letter x.