Find the Volume (Solid of Revolution)

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SUMMARY

The discussion focuses on calculating the volume of the solid formed by revolving the area between the curves y = sin(x) and y = cos(x) around the x-axis, specifically between the limits x = 0 and x = π/4. The volume is determined using the washer method, represented by the integral π * ∫[0, π/4] ((cos(x))^2 - (sin(x))^2) dx. The lower boundary of integration is established at x = 0, where the curves intersect the y-axis. A graphical representation of the region and cross-section is recommended for clarity.

PREREQUISITES
  • Understanding of calculus concepts, particularly integration.
  • Familiarity with the washer method for calculating volumes of revolution.
  • Knowledge of trigonometric functions, specifically sine and cosine.
  • Ability to interpret and sketch graphs of functions.
NEXT STEPS
  • Study the washer method in detail, focusing on its application in volume calculations.
  • Learn how to graph the functions y = sin(x) and y = cos(x) to visualize the area of revolution.
  • Explore examples of volume calculations using cylindrical shells for comparison.
  • Investigate the concept of limits of integration in the context of volume problems.
USEFUL FOR

Students studying calculus, particularly those learning about volumes of solids of revolution, as well as educators seeking to enhance their teaching methods in this area.

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Find the volume of the solid y = sinx, y=cosx, and x= pi/4, revolving around x-axis
I didn't really get this at all... do I plug pi/4 for x in y=sinx, y=cosx to get the integra boundaries?
 
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You need to give us the entire problem, including the axis this region has been revolved around.

After that, show us what you have tried. If you're stuck, your book most likely has some similar examples that show how to calculate volumes of revolution by cylindrical shells or by washers.
 
Mark44 said:
You need to give us the entire problem, including the axis this region has been revolved around.

After that, show us what you have tried. If you're stuck, your book most likely has some similar examples that show how to calculate volumes of revolution by cylindrical shells or by washers.

Sorry, forgot to mention it's revolving around the x-axis. Anyways, I got this from a friend pi * Int[0,pi/4] (cosx)^2 - (sinx)^2 dx), and I was wondering where the lower boundary, 0, came from.
 
The description of the region that will be revolved around the x-axis is not as complete as I would like. I believe it is the region bounded by the graphs of y = sin x and y = cos x between x = 0 (the y-axis) and the line x = pi/4.

If this is the right description, the region is sort of triangular, but with two curved sides. The "base" of this region runs along the y-axis between 0 and 1, and the two curves intersect at (pi/4, sqrt(2)/2).

You should have at least one graph: one showing the region to be revolved, and ideally, another that shows a cross-section of the volume of revolution. Your friend is using washers - disks with a hole in the middle.

The area of a washer is pi*(R^2 - r^2), where R is the larger radius and r is the smaller radius.

The volume of a washer is the area time the thickness, which is pi*(R^2 - r^2)*thickness, which can be either dx (vertical washers) or dy (horizontal washers).

For the limits of integration, figure out where the washers run. Vertical washers run left to right along the interval in question. Horizontal washers run bottom to top along the interval in question.

Can you get started with that?
 

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