This "1/3" Isn't Random... It Finally Makes Sense

Have you ever wondered why the volume of a cone is exactly one-third of the volume of a cylinder having the same base radius and height? Today, in this video, we are going to derive it from scratch using one of the most elegant techniques in calculus called the disk method. The first step is, of course, to draw a cone. Usually, we draw a cone standing upright, with its circular base at the bottom and its tip at the top. The base has radius r and the height of the cone is h. Although this picture looks natural, it is not the easiest one to work with when using calculus. So instead, we rotate the cone sideways. This way, the tip of the cone is at the origin and the circular base is at a horizontal distance of h. Now, do one thing. Draw a straight line joining the tip of the cone to the edge of its circular base. When this line is rotated about the x-axis, it sweeps out the entire cone, right? Our next goal is to find the equation of this straight line. Notice that the line passes through the origin, so its y-intercept is zero. This means there is no constant term. Therefore, all we need is the slope of the line. The slope is simply the vertical rise divided by the horizontal run. From the origin, the line rises by r units while moving horizontally by h units. Therefore, the slope is r divided by h. Its equation becomes y equals r divided by h multiplied by x. Noice! Now consider a very thin vertical strip inside the triangular region. Let its thickness be dx. When this thin rectangle is rotated about the x-axis, it forms a very thin circular disk, right? So, if we imagine adding infinitely many such disks together, they will perfectly fill the entire cone. This is what the disk method is all about. Let us now determine the volume of one such disk. A disk is simply a very thin cylinder. We already know that the volume of a cylinder is the area of its circular face multiplied by its thickness. Here, the radius of the disk is y, so its cross-sectional area is pi multiplied by y squared. Multiplying this by the thickness dx gives the volume dv of one tiny disk. Therefore, in order to obtain the volume of the entire cone v, we simply integrate all these tiny disks, volume dv, from the tip of the cone to the base. Substitute the value of dv as this. Now, the cone starts at the origin where x equals 0 and ends at the circular base where x equals h. Therefore, the integral will be evaluated from 0 to h. However, there is one small problem. Since we are integrating with respect to x, the entire integrand must be written only in terms of x. We cannot leave the variable y inside the integral. So, put y equals r divided by h multiplied by x here. This becomes the integral from 0 to h of pi multiplied by r squared divided by h squared multiplied by x squared multiplied by dx. Before integrating, let us move all the constants like pi, r, and h outside the integral. We are left with pi r squared divided by h squared times the integral from 0 to h of x squared with respect to dx. Using the power rule for integration, the integral of x squared becomes x cubed divided by 3. And we evaluate this expression between the limits 0 and h. Substituting the upper limit gives h cubed divided by 3. Substituting the lower limit gives 0 cubed divided by 3, which is simply 0. Therefore, our expression now becomes h cubed divided by 3. Notice that h cubed divided by h squared leaves just one factor of h. So, finally, we obtain 1 divided by 3 multiplied by pi r squared h. This is exactly the well-known formula for the volume of a cone. That was simply mind-blowing, wasn't it? Like, share, and subscribe. So good.