Find the volume and the surface area of each solid. |...

Name: Problem 23, Chapter 4: Impact Mathematics Algebra and More
Uploaded: 2020-07-08T02:35:12-08:00
Duration: 5 min 38 s
Channel: Noraney Ocampo
Description: Find the volume and the surface area of each solid.

Key Concepts

Calculus-Based Methods

Calculus-based methods are used to find volume and surface area when dealing with irregular or more complex solids. Techniques such as the disk method, washer method, and shell method for volume, as well as surface integrals for surface area, allow the computation of these quantities by breaking the solid down into infinitesimally small elements and summing their contributions. These methods generalize and extend basic geometric formulas to a wider range of shapes.

Volume

Volume is a measure of the three-dimensional space occupied by an object. It quantifies the capacity of a solid and is a fundamental concept in geometry and calculus. Different solids have distinct formulas for calculating volume, and when a direct formula is unavailable, integration methods such as the disk, washer, or shell method can be employed to derive the volume by summing up infinitesimally small elemental volumes.

Surface Area

Surface area refers to the total area of the outer surfaces of a three-dimensional object. It is crucial in many applications, including determining the amount of material needed for coverings or coatings. For common solids, there are standard geometric formulas to compute surface area, but for more complex shapes, calculus methods involving integration over the surface may be necessary.

Solid Geometry

Solid geometry is the branch of mathematics that deals with three-dimensional figures such as cubes, spheres, cylinders, and cones. It focuses on properties like volume and surface area, and understanding the relationships between different dimensions. The principles of solid geometry are applied both in theoretical contexts and practical problem-solving, including the computation of physical quantities in engineering and science.

Transcript

00:01 So you're being asked to find the area, the one surface area and volume.

00:06 So first let's do the volume.

00:09 If you have a cylinder, so you're looking for how much space it takes inside.

00:19 And then the surface area, again, of the cylinder is how much is it taking if you wanted to wrap this up.

00:33 So surface area is wrap.

00:36 So if you're going to wrap it, how much paper you're going to use? and the volume is space it takes, including the inside.

00:49 So how many cubes would it fit inside? so we are given the diameter for this is nine, and the height of this is also nine.

01:00 So i don't really memorize formulas, but except for the basic ones.

01:05 So the area of a circle is high r square.

01:12 So now r is your whole radius.

01:15 In this case, your radius.

01:18 Is 4 .5 because your whole diameter is 9.

01:21 So if you have that circle, you want it nine times.

01:25 So that's one way that i kind of memorized.

01:30 But i didn't memorize, i just memorized the circle.

01:33 So the circles, and you're going to do that nine times.

01:37 So not area, volume.

01:44 So this is my volume formula or a cylinder.

01:48 So in this case, my r, again, is 4 .5, because i only want the radius, i square that and multiply this times pi, times my height is nine.

02:04 So then if i put all this into the calculator, you're going to get 4 .5 squared times 9 times pi...

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