적분의 활용 전체 문제

Consider the region enclosed by the graphs of $y=f(x),y=1+f(x),x=0,y=0,$ and $x=a>0.$ What is the volume of the solid generated when this region is rotated around the $y\text{-axis}?$ Assume that the function is defined over the interval $[0,a].$

A force of $F=20x-{x}^{3}$ N stretches a nonlinear spring by $x$ meters. What work is required to stretch the spring from $x=0$ to $x=2$ m?

[T] For the pyramid in the preceding exercise, assume there were $1000$ workers each working $10$ hours a day, $5$ days a week, $50$ weeks a year. If the workers, on average, lifted 10 100 lb rocks $2$ ft/hr, how long did it take to build the pyramid?

[T] A rectangular dam is $40$ ft high and $60$ ft wide. Assume the weight density of water is 62.5 lbs/ft³. Compute the total force $F$ on the dam when

a. the surface of the water is at the top of the dam and b. the surface of the water is halfway down the dam.

[T] Find the work required to pump all the water out of a cylinder that has a circular base of radius $5$ ft and height $200$ ft. Use the fact that the density of water is $62$ lb/ft³.

[T] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.

[T] How much work is required to pump out a swimming pool if the area of the base is $800$ ft², the water is $4$ ft deep, and the top is $1$ ft above the water level? Assume that the density of water is $62$ lb/ft³.

A cylinder of depth $H$ and cross-sectional area $A$ stands full of water at density $\rho .$ Compute the work to pump all the water to the top.

For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Rotating $y=mx$ around the $x$-axis between $x=0$ and $x=1$

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Rotating $y=mx$ around the $y$-axis between $x=0$ and $x=1$

A general cone created by rotating a triangle with vertices $(0,0),$ $(a,0),$ and $(0,b)$ around the $y$-axis. Does your answer agree with the volume of a cone?

A general cylinder created by rotating a rectangle with vertices $(0,0),$ $(a,0),(0,b),$ and $(a,b)$ around the $y$-axis. Does your answer agree with the volume of a cylinder?

A sphere created by rotating a semicircle with radius $a$ around the $y$-axis. Does your answer agree with the volume of a sphere?

Find the generalized center of mass in the sliver between $y={x}^{a}$ and $y={x}^{b}$ with $a>b.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Find the generalized center of mass between $y={a}^{2}-{x}^{2},$ $x=0,$ and $y=0.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

Find the generalized center of mass between $y=b\,\sin (ax),$ $x=0,$ and $x=\frac{\pi }{a}.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.

In Exercises 13–18, a region of the Cartesian plane is described. Use the Shell Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes.

Region bounded by: $y=\sqrt{x}$, $y=0$ and $x=1$.

Rotate about:

(a) the $y$-axis (b) $x=1$

(a) the $x$-axis (b) $y=1$

In Exercises 13–18, a region of the Cartesian plane is described. Use the Shell Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes.

Region bounded by: $y=4-x^2$ and $y=0$.

Rotate about:

(a) $x=2$ (b) $x=-2$

(a) the $x$-axis (b) $y=4$

In Exercises 13–18, a region of the Cartesian plane is described. Use the Shell Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes.

The triangle with vertices $(1,1)$, $(1,2)$ and $(2,1)$.

Rotate about:

(a) the $y$-axis (b) $x=1$

(a) the $x$-axis (b) $y=2$