# Volumes by Slicing ## Disk ![image-20220928002914467](/home/zack/.config/Typora/typora-user-images/image-20220928002914467.png) ![This figure has four graphs. The first graph, labeled “a” is a parabola f(x)=(x-1)^2+1. The curve is above the x-axis and intersects the y-axis at y=2. Under the curve in the first quadrant is a vertical rectangle starting at the x-axis and stopping at the curve. The second graph, labeled “b” is the same parabola as in the first graph. The rectangle under the parabola from the first graph has been rotated around the x-axis forming a solid disk. The third graph labeled “c” is the same parabola as the first graph. There is a shaded region bounded above by the parabola, to the left by the line x=-1 and to the right by the line x=3, and below by the x-axis. The fourth graph labeled “d” is the same parabola as the first graph. The region from the third graph has been revolved around the x-axis to form a solid.](https://openstax.org/apps/archive/20220815.182343/resources/5d382d402c020cd4b7a6e02c80124d51ade5c47a) ## Washer ![image-20220928003006541](/home/zack/.config/Typora/typora-user-images/image-20220928003006541.png) ![This figure has four graphs. The first graph is labeled “a” and has the two functions f(x)=squareroot(x) and g(x)=1 graphed in the first quadrant. f(x) is an increasing curve starting at the origin and g(x) is a horizontal line at y=1. The curves intersect at the ordered pair (1,1). In between the curves is a shaded rectangle with the bottom on g(x) and the top at f(x). The second graph labeled “b” is the same two curves as the first graph. The shaded rectangle between the curves from the first graph has been rotated around the x-axis to form an open disk or washer. The third graph labeled “a” has the same two curves as the first graph. There is a shaded region between the two curves between where they intersect and a line at x=4. The fourth graph is the same two curves as the first with the region from the third graph rotated around the x-axis forming a solid region with a hollow center. The hollow center is represented on the graph with broken horizontal lines at y=1 and y=-1.](https://openstax.org/apps/archive/20220815.182343/resources/e677e3390d28925844052b6e3bccde3accc53315) ### Example ![image-20220928003210155](/home/zack/.config/Typora/typora-user-images/image-20220928003210155.png) ![image-20220928003220856](/home/zack/.config/Typora/typora-user-images/image-20220928003220856.png) # Volumes by Shells ![image-20220928003302243](/home/zack/.config/Typora/typora-user-images/image-20220928003302243.png) ![This figure has two images. The first is a cylindrical shell, hollow in the middle. It has a vertical axis in the center. There is also a curve that meets the top of the cylinder. The second image is a set of concentric cylinders, one inside of the other forming a nesting of cylinders.](https://openstax.org/apps/archive/20220815.182343/resources/e2c54b488957e854471497d747225838ced711a2) # Arc Lengths ![image-20220928003755108](/home/zack/.config/Typora/typora-user-images/image-20220928003755108.png) ![This figure is a graph. It is a curve above the x-axis beginning at the point f(xsubi-1). The curve ends in the first quadrant at the point f(xsubi). Between the two points on the curve is a line segment. A right triangle is formed with this line segment as the hypotenuse, a horizontal segment with length delta x, and a vertical line segment with length delta y.](https://openstax.org/apps/archive/20220815.182343/resources/69c06ea19e9c7d18e50d6ce97112b3eaac9c7eff) ![image-20220928003852525](/home/zack/.config/Typora/typora-user-images/image-20220928003852525.png) # Physical Applications ![image-20220928004027547](/home/zack/.config/Typora/typora-user-images/image-20220928004027547.png) ![image-20220928004306876](/home/zack/.config/Typora/typora-user-images/image-20220928004306876.png) ![image-20220928004501373](/home/zack/.config/Typora/typora-user-images/image-20220928004501373.png) ![image-20220928004524116](/home/zack/.config/Typora/typora-user-images/image-20220928004524116.png) ![image-20220928004616696](/home/zack/.config/Typora/typora-user-images/image-20220928004616696.png) ![This image is a graph of y=f(x). It is in the first quadrant. Under the curve is a shaded region labeled “R”. The shaded region is bounded to the left at x=a and to the right at x=b.](https://openstax.org/apps/archive/20220815.182343/resources/df3152018c89d6a7aaafdb051ee8eb6d4569925e) ![This figure is a graph of the curve labeled f(x). It is in the first quadrant. Under the curve and above the x-axis there is a vertical shaded rectangle. the height of the rectangle is labeled f(xsubi). Also, xsubi = xsubi – 1 + xsubi/2.](https://openstax.org/apps/archive/20220815.182343/resources/4edea7bac957c4e910e44d8cf6308c13826cee17) ![image-20220928004920818](/home/zack/.config/Typora/typora-user-images/image-20220928004920818.png) ![image-20220928004937100](/home/zack/.config/Typora/typora-user-images/image-20220928004937100.png) ![image-20220928004950084](/home/zack/.config/Typora/typora-user-images/image-20220928004950084.png)