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- 4a. Volume of Solid of Revolution by Integration (Disk method)
- 6.4: Arc Length of a Curve and Surface Area
- Appropriate Integrals
- Arc Length of the Curve [latex]y[/latex] = [latex]f[/latex]([latex]x[/latex])

AP Calculus BC. Search this site. General Resources.

AP Calculus BC. Search this site. General Resources. Summer Assignments. Tuesday, July 10th. Wednesday, Week 1. Week 1 - Friday. Week 1 - Monday. Week 1 - Thursday. Week 2 - Friday. Week 2 - Monday. Week 2 - Thursday. Week 2 - Tuesday. Week 2 - Wednesday. Week 3 - Monday. Week 3 - Thursday. Week 3 - Tuesday. Week 3 - Wednesday. Semester 2 - Week S1 - Week 1. Semester 1 - Week 2. Semester 1 - Week 3. Semester 1 - Week 4. Semester 1 - Week 5.

Semester 1 - Week Semester 1 - Week 6. Semester 1 - Week 7. Semester 1 - Week 8. Semester 1 - Week 9. Semester 2 - Week 1. Semester 2 - Week 2. Semester 2 - Week 3. Semester 2 - Week 4. Semester 2 - Week 5. Semester 2 - Week 6. Semester 2 - Week 7. Semester 2 - Week 8. Semester 2 - Week 9. Monday The warm-up was a review of the Shell Method of calculating volumes. We calculated the volume of a torus. Try it! We then worked on a group investigation to determine a formula for the arc length of any curve.

Note the similarities and differences between the two formulas. The surface area formula above is for an object rotated around the x-axis. There are possible variations for which we have to modify the integral:. The only change necessary is the "radius" - it changes to simply being x :. It's best if you remember where the formulas come from to figure out the "radius" and the "dL" the square root part in each formula. See me for help with the formulas if needed.

That was a straightforward application of the arc length formula:. We then went through the process of finding the surface area obtained by revolving that arc around the y-axis. Yesterday, I went through an example of rotating around the y-axis in which we switched everything to a dy integral.

That is fine, but it's actually not necessary. If you revolve around the y-axis, the radius of your rotation is simply x. If you revolve around the x-axis, the radius of rotation is y, or f x. We then began to review for the unit test on Friday. Topics for the test:. For the test, you are allowed one half-sheet of hand-written notes of your choosing to help you remember stuff. You can put anything you want on the notes, but it must be in YOUR handwriting. More review. The warm-up was a review of finding areas between curves.

Make sure you know how to find intersection points and how to graph common functions. We then summarized the three methods for computing volumes. All methods are based on the idea of volume as an integral of section areas multiplied by a dx thickness. Constant cross-sections - The A x function depends on the wording in the problem - squares, semi-circles, triangles, etc. Know your geometry.

They may have a hole in the middle. For dx-integrals, the radius, r, is the distance from the axis of rotation to the x-coordinate and the height is f x.

My apologies if there are errors. I did it quickly, but it still took seven pages of work and almost two hours!

Screen Shot at 1. Screen Shot at 2. Warm-ups 37 - Shell method. The homework was out of the book: WI p. There is no need to convert to dy unless it makes the problem easier. Wednesday More review.

Many solid objects, especially those made on a lathe , have a circular cross-section and curved sides. On this page, we see how to find the volume of such objects using integration. NOTE: On this page we use the disk method and washer method where we cut the shape into circular slices only, and meet the Shell Method next. When we rotate such a shape around an axis, and take slices, the result is a washer shape with a round hole in the middle. Find the volume of the material needed to make the cup.

Area of a Surface of Revolution The area between the curve and the x axis is the definite integral. The new applications to volume and length and surface.

In mathematics , an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation , integration is a fundamental operation of calculus, [a] and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals , which can be interpreted formally as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I've noticed the following things while studying calculus, and would like experts to tell me if my conclusions are right. Why or why not? This observation holds for all balls, cubes, and simplexes provided they're centered at the origin.

In this section, we use definite integrals to find the arc length of a curve. We can think of arc length as the distance you would travel if you were walking along the path of the curve. Many real-world applications involve arc length. If a rocket is launched along a parabolic path, we might want to know how far the rocket travels.

This section contains documents created from scanned original files, which are inaccessible to screen reader software. A " " symbol is used to denote such documents. Method for finding the area between two curves. Introduces the disk method with worked examples of finding the volume of a right circular cone and a sphere. Variation of disk method using the difference of two disks to create washers.

In this section we are going to look at computing the arc length of a function. We can then approximate the curve by a series of straight lines connecting the points. In other words, the exact length will be,. We can then compute directly the length of the line segments as follows. However, using the definition of the definite integral , this is nothing more than,. This formula is,.

Applications to Area, Arc length, Volume and Surface area. Suppose f(x) ≥ 0 on [a, b]. Then it is clear from the definition of Definite integral that the area.

In this section, we use definite integrals to find the arc length of a curve. We can think of arc length as the distance you would travel if you were walking along the path of the curve. Many real-world applications involve arc length.

You can find the volume of an irregular object by immersing it in water in a beaker or other container with volume markings, and by seeing how much the level goes up. Solid of revolution between two functions leading up to the washer method. In about , the Flemish chemist Jan Baptist van Helmont observed that when he burned charcoal in a closed vessel, the mass of the resulting ash was much less than that of the original charcoal.

Ему захотелось увидеть ее глаза, он надеялся найти в них избавление. Но в них была только смерть. Смерть ее веры в. Любовь и честь были забыты.

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