以下均为学习中未能解决的问题，务必多加思索，找到解法。

1.Find the volume of the solid of revolution generated when the area bounded by the given curves is revolved about the x-axis:$x=2y-y^{2}$ , $x=0$   产生于2023.10.09

2.Two great circles lying in planes that are perpendicular to each other are drawn on a wooden sphere of radius $a$. Part of the sphere is then shaved off in such a way that each cross section of the remaining solid that is per­pendicular to the common diameter of the two great circles is a square whose vertices lie on these circles. Find the volume of this solid. 产生于2023.10.14

3.PROBLEMS OF 7.7 WORK AND ENERGY (Calculus With Analytic Geometry page251)-problems 26&27 (Newton's and Einstein's theories) 产生于2023.11.4

4.Two oblique circular cylinders of equal height $h$ have a circle of radius $a$ as a common lower base and their up­per bases are tangent to each other. Find the common volume. 产生于2023.11.11（两圆相交面积无法建立公式）

5.Consider the torus generated by revolving the circle $\left( x-b\right) ^{2}+y^{2}=a^{2}( 0 <a < b)$ about the y-axis. Use the shell method to show that the volume of this torus equals the area of the circle times the distance traveled by its center during the revolution. Hint: At the right moment, change the variable of integration from $x$ to $z=x-b$. 产生于2023.11.11

6.Consider a falling body of mass $m$ and assume that the retarding force due to air resistance is proportional to the square of the velocity. If the body falls from rest, find a formula for the velocity in terms of the distance fallen, and thereby find the terminal velocity in this case. Hint: $\dfrac{dv}{dt}=\dfrac{dv}{ds}\dfrac{ds}{dt}=v\cdot \dfrac{dv}{ds}$. 产生于2023.12.24

7.If one side and the opposite angle of a triangle are fixed, and the other two sides are variable, use the law of cosines to show that the area is a maximum when the triangle is isosceles. (Can you prove this by using geometry alone?). 产生于2024.1.2

8.A heavy spherical ball is lowered carefully into a full conical wine glass whose depth is $a$ and whose gener­ating angle (between the axis and a generator) is $\alpha $. Show that the greatest overflow occurs when the radius of the ball is $$\dfrac{a\sin \alpha }{\sin \alpha +\cos 2\alpha }$$ 产生于2024.1.3

9.A rope with a ring at one end is looped over two pegs in a horizontal line.The free end is passed through the ring and has a weight suspended from it,so that the rope is held taut. If the rope slips freely through the ring and over the pegs, then the weight will descend as far as possible in order to minimize its potential energy. Find the angle formed at the bottom of the loop. 产生于2024.1.14（题目都看不懂，无语）

10.The circle $x^{2}+y^{2}=a^{2}$ is revolved about the line $y=a$. Find the area of the resulting surface of revolution. 产生于2024.1.21

11.Let the pendulum in Example 3 be pulled to one side through an angle $\alpha$ and released. Use the principle of conservation of energy to show that the period $T$ of oscillation is given by the formula$$T=4\sqrt{\dfrac{\alpha }{2g}}\int _{0}^{\alpha }\dfrac{d\theta }{\sqrt{\cos \theta -\cos \alpha }}$$ 产生于2024.2.16（page333）