Calculating Volumes via Disks or Washers
Read this topicExplanations 8
- To find the area under a given curve, we've previously seen that one can consider a series of rectangles with constant width inscribed under a curve, and consider their total ar…
By David Stoner, Ahaan Rungta, Alex Federspiel
- Volume calculated with disks where A(x)=π[f(x)]2A(x)=\pi[f(x)]^2A(x)=π[f(x)]2 if fff is rotated around the xxx-axis, or A(x)=π[a−f(x)]2A(x)=\pi[a−f(x)]^2A(x)=π[a−f(x)]2 if fff i…
By Yumeng Zhu, Japheth Wood
- The volume of a solid is can be found using the definite integral ∫abA(x)dx\int_a^{b} A(x)dx∫abA(x)dx). One method for using a definite integral to compute the volume bounded b…
By Noah Libby
- Lets go over the disk method. The disk method is a method using integration and a little geometry wit to find the volume of a rotated 2-D figure. This has application in many fi…
By Michael U, Alex Federspiel
- (Video) Using Disks or Washers to Find Volume by PatrickJMT In this video by patrickJMT he explains the disk or washer method of finding the volume of a region. Suppose you have…
By Alex Federspiel
- (Videos) Volumes of Revolution Using Disks and Washers by PatrickJMT Here's a video by patrickJMT where he works through some more examples of finding the volume of a shape rota…
By Alex Federspiel
- (Video) Volumes of Revolution about Vertical Lines Using Washers by PatrickJMT In this video by patrickJMT he shows you how to find the volume of a revolution about a vertical l…
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 15
- Rounding to the nearest percent, what fraction of the Earth's volume lies above the 303030 degrees north parallel? You may assume the Earth is a perfect sphere. The volume in qu…
- Find the volume of the solid formed when we rotate the region bounded by y=x−1y = \sqrt{x-1}y=x−1, y=0y=0y=0, and x=5x=5x=5 about the xxx-axis. Round to the nearest tenth.
- Calculate the volume of a truncated right circular cone (conical frustum) whose base radius is 4, top surface radius is 2, and height is 3. (Please round to an integer number.)
- Calculate the volume of a solid created by rotating the curve y=sin32x (0≤x≤π)y=\sin^{\frac{3}{2}}x~(0\leq x\leq\pi)y=sin23x (0≤x≤π) about the xxx-axis.
- A unit radius ball (sphere with filled interior) is divided, by 2 parallel planes, into 3 solids. Suppose that for each solid, the ratio of the surface area to volume has the sa…
- Find the volume of the region bounded by y=1−x2y=1-x^2y=1−x2 and y=0y=0y=0 and rotated about the xxx-axis.
- Find the volume of the region bounded by y=lnxy=\ln xy=lnx, y=0y=0y=0, y=2y=2y=2, and x=0x=0x=0 and rotated about the yyy-axis. Round to the nearest integer.
- Find the volume of the region bounded by y=4x2y=4x^2y=4x2 and y=5−x2y=5-x^2y=5−x2, and rotated about the xxx-axis.
- Find the volume of the region bounded by y=xy=xy=x and y=2xy=2\sqrt xy=2x, and rotated about the line y=4y=4y=4.
- Find the volume of the region bounded by x=1−y2x=\sqrt{1-y^2}x=1−y2 and x=−1−y2x=-\sqrt{1-y^2}x=−1−y2, and rotated about the line x=−2x=-2x=−2.
- Gabriel's Horn is defined as the shape generated when we revolve the graph of y=1xy = \frac{1}{x}y=x1 around the xxx-axis for x≥1.x \geq 1.x≥1. What is the volume of Gabriel's …
- We define Gabriel's Modified Horn as the shape generated when we revolve the graph of y=1xαy = \frac{1}{x^{\alpha}}y=xα1 around the xxx-axis for x≥1,x \geq 1,x≥1, where α>12\al…
- We define Gabriel's Modified Horn as the shape generated when we revolve y=1xαy = \frac{1}{x^{\alpha}}y=xα1 around the xxx-axis for x≥1,x \geq 1,x≥1, where α>12\alpha > \frac{1…
- We define Gabriel's Modified Horn as the shape generated when we revolve y=1xαy = \frac{1}{x^{\alpha}}y=xα1 around the xxx-axis for x≥1,x \geq 1,x≥1, where α>12\alpha > \frac{1…
- We define Gabriel's Modified Horn as the shape generated when we revolve the graph of y=1xαy = \frac{1}{x^{\alpha}}y=xα1 around the xxx-axis for x≥1,x \geq 1,x≥1, where α>12\al…