Find the volume of a pyramid with height h and rectangular base with dimensions b and 2b.
Essentially, I'm trying to derive the volume of a pyramid using integrals and a graph and revolving the plane about an axis. I hope that's kind of clear and you guys understand what I'm asking.
Thank you very much
That seems to me to be the hard way to integrate.
take a small rectangular box dV somewhere along the height.
dV= dh*darea
darea= dw dl=(H-h)(b)*(H-h)2b /H^2
dV=1/H^2 (H-h)^2 2b^2 dh h from 0 to H
V= 2b^2/H^2 INT (H-h)^2 dh= 2b^2 /3 H^2 *(H-h)^3 from zero to H
=2b^2/3 H
which you knew, the volume of any rectangular prism is 1/3 area base*height
To find the volume of a pyramid with a rectangular base, we can use integration.
First, let's start by setting up our coordinate system where the base of the pyramid lies in the xy-plane. The base has dimensions b and 2b, so the coordinates of the four corners are (-b, -b), (-b, b), (b, b), and (b, -b).
Now, let's imagine a line segment parallel to the y-axis from the point (x, y) on the base to the top of the pyramid. As we revolve this line segment around the y-axis, it will create a disk.
The radius of this disk changes as we move up the pyramid. For a given y-coordinate, the radius can be found using similar triangles. The height of the segment corresponding to a given y-coordinate is h - y, and the corresponding side length on the base is 2b - (2b/b) * y = 2b - 2y.
Thus, the ratio of the radius (r) to the height of the segment (h - y) is the same as the ratio of the corresponding side length on the base (2b - 2y) to the height of the base (2b).
Setting up this proportion, we get:
(r)/(h - y) = (2b - 2y)/(2b)
Cross-multiplying and simplifying, we have:
r = (h - y)(2b)/(2b - 2y)
Now, we can find the area of each individual disk using the formula for the area of a circle (pi * r^2), where r is given by the equation above.
Let's integrate the area of each disk from the bottom of the pyramid (y = -b) to the top (y = b) to find the total volume of the pyramid.
The volume (V) can be found by integrating the area (A) with respect to y:
V = ∫[from -b to b] A(y) dy
= ∫[from -b to b] (pi * (h - y)^2 * (2b)^2 / (2b - 2y)^2) dy
Simplifying the equation, we have:
V = pi * (2b)^2 * ∫[from -b to b] (h - y)^2 / (2b - 2y)^2 dy
Evaluating this integral will give us the volume (V) of the pyramid with height h and rectangular base dimensions b and 2b.
To find the volume of a pyramid with height h and rectangular base with dimensions b and 2b, you can use integration and the concept of solids of revolution. Here's how you can derive the formula:
1. Start by sketching the pyramid. The base will be a rectangle with dimensions b and 2b, and the height of the pyramid will be h.
2. Now, consider a cross-section of the pyramid, perpendicular to the base. This cross-section will be a rectangle, with width b and height y, where y varies from 0 to h.
3. To find the area of this cross-section, you multiply the width b by the height y, giving you b * y.
4. Now, imagine revolving this rectangle around one of its sides, creating a solid. The resulting solid will be a cone, with a radius equal to the width of the rectangle (b) and a height equal to the height of the pyramid (h).
5. The volume of a cone can be expressed as V = (1/3) * π * r^2 * h, where r is the radius. In this case, the radius is b.
6. To find the volume of the pyramid, we need to subtract the volume of the pyramid below the cross-section from the volume of the cone. The volume of the pyramid below the cross-section is given by (1/3) * b * y * y, where y is the height of the cross-section.
7. Integrate the difference in volumes from y = 0 to y = h:
∫ [(1/3) * π * b^2 * h - (1/3) * b * y^2] dy
Integrating this expression will give you the volume of the pyramid.
It's worth noting that if you just need the formula for the volume of a pyramid and don't necessarily want to derive it using calculus, the general formula is V = (1/3) * base area * height. For a rectangular pyramid, the base area is given by b * 2b = 2b^2, so the volume simplifies to V = (1/3) * 2b^2 * h = (2/3) * b^2 * h.