in this video we're going to talk about how to calculate the volume by rotating a region around the x-axis or the y-axis using the disc method and the washers method so let's begin let's say if we have a curve and the curve looks like this and we wish to calculate the volume of the curve from let's say a to b one method that we can use is a disc method and the disc method works by taking a cross-sectional area this is going to be the radius but if we take the cross-sectional area we can turn it into a disc as we rotate the region about the x-axis we could form basically something that looks like a disk now i'm going to draw this disk the other way notice that this forms the shape of a cylinder the cylinder has a radius r and it has a height h the volume of a cylinder is basically the volume of the cross-sectional area times the height the cross sectional area being the circle on top which is pi r squared thus the volume of the cylinder is prior square times height and so the volume of this blue disc it has a height h which you can also view it as a dx or delta x and it has a cross sectional area which is pi r squared now to find the volume of a solid we need to add up all the little discs from a to b if you make other disc there's another disc that you can make my drone is kind of terrible but basically the idea is that you add up all these smaller discs to find the total volume of the uh of the object or the solid that is created by revolving about the x-axis but to find the volume you got to find a volume of you have to integrate the cross-sectional area from a to b and the area is basically pi r squared where r is going to be a function of x and so this is the equation to find the volume using the disc method when revolving about the x-axis the main idea needs to understand is that the cross-sections that are used for this technique is basically a circle and the area of a circle is pi r squared now sometimes you may need to find the area when it's revolved or rotated about the y axis but let's just go over some basic things so if you want to find the volume rotate about the x-axis this is going to be the shader region and it has to be in terms of x so the radius is between the x-axis that's where the solid is being rotated about and it's between the curve so that's the radius in terms of x so the volume is going to be from you have to integrate the function from a to b and it's going to be r squared of x dx that's how you find the volume of the solid rotated about the x-axis now let's say if you want to find it about the y axis instead of using a and b we're going to use c and d which represent y values the radius is going to be between the y-axis and the curve since we're rotating about the y-axis so the volume is going to be pi integration from c to d r of y squared times d y so that's how you find the volume rotate about the y axis or a line parallel to the y axis and this is for the x-axis or line parallel to the x-axis all right let's start with an example let's say we have the function y is equal to the square root of x and we want to find the volume when rotating this curve about the x-axis and we want the portion of the curve where it's bounded between x is equal to zero and x is equal to four so feel free to try this problem the first thing you want to do is you want to plot the function the square root of x looks like this the radius is the distance between the curve and the axis of rotation in this case we're rotating about the x-axis and we want the portion of the curve between x is equal to zero and x is equal to four so this is the region that we want the first thing you want to do is find out what r of x is equal to r of x is equal to y which y is equal to square root x so r of x is equal to both of these things but we want it to be in terms of x so we're going to use square root x now let's use the equation the volume is equal to pi times the integration from a to b r of x and dx and let's not forget to uh square it so this is going to be pi a is 0 b is 4 r of x is the square root of x and we're going to square that and then we have dx root x squared is basically x so let's integrate this function the antiderivative of x is going to be x squared over 2. you have to add one to the exponent and then divide by that result so now let's find the value of this uh definite integral so first let's plug in four so it's going to be four squared over two and then we'll plug in zero four squared is sixteen zero squared is just zero and uh sixteen divided by two is eight so the final answer is eight pi now it's your turn find the volume of the solid generated by rotating the function y is equal to one over x about the x-axis bounded by the region x is equal to one and the line x is equal to three so feel free to pause the video and work on this example use the disk method to calculate the volume so the first thing we need to do is plot the function so the right side of 1 over x it looks like this but we only want the portion from one to three so we only want this region and we're going to rotate it about the x-axis so the radius r of x is the distance between the x-axis or the axis of rotation and the curve so r of x is simply equal to the function one of x the way you find it is you take the top part of the function or the top part of this line which is one over x minus the bottom part which is uh y is equal to zero that's the x axis so it's one over x minus zero which is just one of x now even though that process seems useless at this point it's useful when you're rotating about an axis that is neither the x or y axis which we'll cover later in this video but for now r of x is simply one of x so the volume is going to be pi integration from a to b or one to three and then it's going to be r of x squared dx so this is pi one two three one over x squared dx one over x squared we need to rewrite it we can write it as x to negative two and now let's use the power rule let's add one to the exponent negative two plus one is negative one and then we need to divide by negative 1. so rewriting the function we can bring the x back to the bottom to get rid of the negative exponent so this is going to be pi times negative one over x evaluated from one to three so this is going to be pi times negative one over three minus negative one over one always start with the top number three and then subtracted by the one on the bottom so this is basically negative one third plus one over one now we need to get common denominators so let's multiply this by three over three so what we have is negative one over three plus three over three negative one plus three is two so the final answer is just two pi over three now let's try a few examples of finding the volume of a solid when it's rotating about the y axis so let's say that y is equal to x squared and the curve is bounded by the lines x equals zero and y equals four and we're rotating about the y axis so let's begin by drawing a graph so this is the y-axis which we're rotating about and the graph y equals x squared the right side of it looks like this now this is the line x equals zero it's basically the y axis and we have the line y equals four which is basically a vertical line so this is the shaded region that's the region that we're interested in the radius is between the axis of rotation which is the y-axis and the curve so that is basically that's r of y now notice that r of y is the same as x x is the distance between the y axis and the curve so we could say that r of y is equal to x now we need to get r of y in terms of y if we want to find the volume rotate about the y axis so we need to find out what x is equal to in terms of y so we have this equation if y is equal to x squared then if we take the square root of both sides we can see that x is equal to the square root of y so let's replace x with root y so the radius in terms of y is equal to the square root of y now to find the volume of this region when it's rotated about the y axis we can use this equation v is equal to pi integration from c to d those are values on the y axis r of y d y squared at the origin the y value is zero and at the line y equals four y is four so therefore c is zero d is four r of y we know it's the square root of y and we have to square it the square root of y squared is just going to be y so now we can integrate the function the integration of y to the first power is y squared over two evaluate it from zero to four and let's not forget the constant in front so now let's plug in those numbers let's start with the top number four so it's four squared over two minus zero squared over two four squared is sixteen zero squared is just zero and sixteen divided by two is eight so the volume is going to be 8 pi now it's your turn try this example let's say y is equal to x raised to the 2 thirds and the curve is bounded by the lines x equals zero and y equals one and we're going to rotate it about the y axis find the volume of this solid that forms once you rotate this curve bounded by those lines about the y axis feel free to pause the video as you work on this example so let's begin with a graph so the graph y equals x to the two thirds it's an increase in function and it increases a decrease in rate it looks like that the line x equals zero is basically the x axis and y equals one that's a horizontal line so we can see that the y values are zero and one and we're rotating about the y axis so this is the shaded region and the radius that's r of y now once again r of y is equal to x and it turns out that r of x is equal to y now let me explain so let's say if we have this curve and we know this is x x is the distance between the y axis and the curve and that's also the radius relative to the y-axis so r of y is always going to be equal to x now the first example when we had y equals x squared and we were revolving about the x-axis r of x was the distance between the x-axis and the curve but notice that r of x is parallel to the y-axis so r of x is equivalent to y but since y is equal to x squared and if r of x is equal to y r of x is equal to x squared and that's what we did in the first example but it's important to understand that r of x is equal to y and r of y is equal to x now let's go back to this problem so if r of y is equal to x what's r of y in terms of y so now let's use the equation that we have in the beginning so we know y is equal to x raised to the two thirds so we need to solve for x and the only way we can do that is by raising both sides to three over two which is the reciprocal of two over three when you raise one exponent to another you need to multiply the two exponents two over three times three over two is one the threes cancel and the twos will cancel so you get x to the first power or simply x therefore x is y to the three over two and since r of y is equal to x then r of y is also equal to y raised to the three over two all we need to do is replace x with y to the three halves now once you have the radius in terms of y we could find the volume of the solid that is produced when rotating about the y axis so now let's use this formula so let's replace c with zero and d with one and let's replace r y with y to the three halves and let's not forget to square it y to the three-halves squared we need to multiply the two exponents three over two times two which is two over one the twos cancel so you just get three so what we now have is v is equal to pi integration zero to one y to the third d y the anti-derivative of y to the third is y to the fourth over four evaluated from zero to one multiplied by pi so let's begin by plugging in one so it's one to the fourth over four minus zero to the fourth over four one to the fourth is basically one zero to the fourth is just zero so the final answer is pi over four and that's it for this problem