![]() ![]() Saying that the function that F of x or y is equal to F We just have to re-express x plus y is equal to one. Let's say that right over here at x, what is going to be theĭiameter of the disc at x. Let's just take theĪpproximating case first. Then, if you took the limitĪs you get an infinite number of discs that are infinitely thin, then you're going to get the exact volume. Of each of those discs, and sum them up then that would be a pretty good approximation for the volume of the whole thing. Let's split up the figure into a bunch of discs. Is, well let's just think about each of these, This three dimensional figure that we are attempting to visualize. You, I encourage you to pause the video and see if you can figure out the volume of this thing that Section right over here, along the y axis, that It would be thatĬross-section right over here, which is a semi-circle. Looking at it at an angle, and if the figure were transparent, it would be thisĬross-section right over here. I've laid the coordinate plane down flat and I'm looking at it from above. If we take a slightly different view of the same three-dimensional, we would see something like this. Sections I guess we could say that are parallel to the yĪxis, so let's say we were taking a cross-section like that, we know that this will be a semi-circle. Sections that are perpendicular to the x axis, so cross Three dimensional figure is if we were to take cross So this region right over here is the base of a three dimensional figure. Quadrant, that this is the base of a three dimensional figure. That's below this graph but still in the first Over here is the graph of x plus y is equal to one. Substituting numbers back into original equation:Ĭan someone please tell me if this method is wrong and if so, what is my error? Again, since the formula for f(x) is x+y=1, I figured the base diameter had to be 1. To calculate the radius, I noticed that another base of the triangle formed by the function f(x) in the first quadrant was the diameter of the cone. Since the formula for f(x) is x+y=1, I figured the base had to be 1. To get the height, I saw that in the diagram of the cone, you can see that the height of the cone is the same as the one of the bases of the triangle formed by the function f(x) in the first quadrant. From here on, I only need to substitute in the values for r and h to get the correct answer. To get half a cone, I divided that by 2, giving me the formula ((pi*r^2*h)/3)/2. ![]() The volume to calculate the volume of a cone is (pi*r^2*h)/3. ![]() I first recognized the shape to be half a cone. Can anyone please tell me if the method I used (shown below) is wrong? Integrating curvatures over beam length, the deflection, at some point along x-axis, should also be reversely proportional to I.1:20, when Sal encouraged us to figure out the volume of the shape, I used a different method that still gave me the correct answer. Therefore, it can be seen from the former equation, that when a certain bending moment M is applied to a beam cross-section, the developed curvature is reversely proportional to the moment of inertia I. Thus their combined moment of inertia is: These triangles, have common base equal to h, and heights b1 and b2 respectively. The moment of inertia of a triangle with respect to an axis perpendicular to its base, can be found, considering that axis y'-y' in the figure below, divides the original triangle into two right ones, A and B. This can be proved by application of the Parallel Axes Theorem (see below) considering that triangle centroid is located at a distance equal to h/3 from base. The moment of inertia of a triangle with respect to an axis passing through its base, is given by the following expression: Where b is the base width, and specifically the triangle side parallel to the axis, and h is the triangle height (perpendicular to the axis and the base). The moment of inertia of a triangle with respect to an axis passing through its centroid, parallel to its base, is given by the following expression: ![]()
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