Calculating Volumes via Cylindrical Shells
Read this topicExplanations 4
- Previously, we calculated areas through thin slices. Volume can be calculated in the same way! If we take thin slices whose volume we can calculate, we can integrate to find the…
By Alexander Mason, Alex Federspiel
- If we want to find the volume of the function pictured above, y=−(x−4)2+1y=-(x-4)^2+1y=−(x−4)2+1 from x=3x=3x=3 to x=5x=5x=5 when rotated around x=6x=6x=6, we run into some prob…
By Lily Zacharias, Japheth Wood, Alex Federspiel
- (Video) Volumes using Cylindrical Shells by PatrickJMT In this video by patrickJMT he shows you how you can use cylindrical shells in order to find the volume of a shape. This a…
By Alex Federspiel
- To get a better understanding of this, please check out the video explanations by patrickJMT for disks/washers and shells. It's important to note that you can use either method …
By Alex Federspiel
Problems 10
- Let TTT be a triangle with vertices at (1003,0)(1003,0)(1003,0), (1004,3)(1004,3)(1004,3), and (1005,1)(1005, 1)(1005,1) in the xyxyxy-plane. Let VVV be the volume of the solid …
- Find the volume of the region bounded by y=x2y=x^2y=x2, y=0y=0y=0, and x=1x=1x=1 about the yyy-axis.
- Find the volume of the region bounded by y=4−2xy=4-2xy=4−2x and y=4+x−2x2y=4+x-2x^2y=4+x−2x2 about the yyy-axis.\\Round your answer to the nearest tenth.
- Find the volume of the region bounded by y=x+3y=x+3y=x+3 and x=y2−6y+9x=y^2-6y+9x=y2−6y+9 about the xxx-axis.
- Find the volume of the region bounded by y=−x2y=-x^2y=−x2, y=1y=1y=1, x=1x=1x=1, and x=2x=2x=2, rotated about the line x=1x=1x=1.
- Find the volume of the region bounded by y=4x−x2y=4x-x^2y=4x−x2 and y=8x−3x2y=8x-3x^2y=8x−3x2 and rotated about x=−2x=-2x=−2.
- Find the volume of the region bounded by y=x+5xy=x+\frac5xy=x+x5 and y=6y=6y=6 and rotated around the line x=6x=6x=6. \\ Round your answer to the nearest integer.
- Find the volume of the region bounded by y=x+1y=\sqrt{x+1}y=x+1, y=0y=0y=0, and x=3x=3x=3 and rotated around the line x=4x=4x=4.
- Find the volume of the region bounded by y=x3y=x^3y=x3, x=y3x=y^3x=y3, y≥0y\ge0y≥0, and x≥0x\ge0x≥0 and rotated around the line y=−1y=-1y=−1.
- Find the volume of the region bounded by x=y2−72yx=y^2-\frac72yx=y2−27y and x=yx=\sqrt yx=y and rotated around the line y=1829/360y=1829/360y=1829/360.