hey guys in this video we're gonna find the volume using the dishwasher method and the cylindrical shell method so what's the difference and how do we know which one to use well don't worry it's very easy so let's start with the dishwasher method here we have a function y is equal to x to the power of 3 and we want to take the area between the y-axis and this function so we want to take the area here and rotate it around the y-axis so as you can see this is the three-dimensional object that we want to find the volume of so if the object looks like half of a cucumber then you have to use the dishwashing method and this method says that if the cross-section is on the y-axis then you integrate with respect to y if it's on the x-axis like this then you have to integrate with respect to x since it's on the y-axis the volume formula is going to be equal to the integral from a to b where a is the bottom right here and b is a very top so b is right here of a of y d y where a of y is the area of the cross section now i'm going to show you an easy trick to remember this formula all you have to do is understand it so basically this part is saying that we're going to find the area of the cross sections from a to b so starting from a we're going to find the area of the cross section here and then we're going to find the area of the cross section here and then here and then here and i think you kind of get the idea so we're gonna keep finding the area of the cross sections until we reach the very top where d of y is the thickness of this dish so it's like the thickness the distance from here to here that's going to be the thickness of the dish and at the end of the day the integral sign means that we're gonna add all of the areas of the cross sections from the top to the bottom and that's gonna give us the volume of this three-dimensional object let's go ahead and find the volume so step number one we want to find a and b so since a is right here at the center then a is equal to zero where b is equal to eight so i wrote the number eight right there so b is going to be equal to eight step number two is to find the area of the cross section so since the cross section or the dish is a circle then the area must be pi times the radius to the power of two so the area a of y is equal to pi times the radius to the power of two now the distance from the center until we reach the cross section is y and the distance from the y axis until we touch this function this distance is x and so the radius is x so the area is the same as pi times x to the power of 2. so since we're trying to integrate with respect to y we need to rewrite this function in terms of y now we know that y is equal to x to the power of three so if we take the cube root of both sides we're gonna get x is equal to y to the power of one third and we can replace it with x right here so finally the area is equal to pi times y to the power of 1 3 to the power of 2 and that will simply give us pi times y to the power of two thirds now we found the area of the cross section we can go ahead and substitute it back into the integral the third and final step is to evaluate this integral so since pi is just a constant we can move it outside of the integral so v is equal to pi times the integral from 0 to 8 of y to the power of 2 over 3 d y this is the same as pi times the anti-derivative of y to the power of two over three so we're going to get y to the power of five over three and then we divide by five over three so when you divide by a fraction it's the same thing as multiplying by its reciprocal and then the limit goes from 8 to 0. so the next step is just substituting these numbers into our formula we're going to get pi times 3 over 5 times 8 to the power of 5 over 3 minus 3 over 5 times 0 to the power of five over three so we know right away this right here is going to be zero now what is a to the power of five over three well a to the power of 5 over 3 is the same as 8 to the power 1 over 3 to the power of 5 which is the same as 2 to the power of 5 which is just 32. so pi times 32 times 3 is going to be 96 over 5 and the final answer is equal to 96 over 5 times pi now you can leave it the way it is or you can say union to the power of 3. basically that means we have found the volume of this three-dimensional shape using the dishwasher method now let's take a look at the second method to find the volume which is the cylindrical shell method so we have a function y is equal to two times x to the power of two minus x to the power of three and we're gonna take the area between this function and the x-axis and rotate it around the y-axis before i show you how to find the volume i would like you to pause the video and try finding the volume using the dishwasher method and you will notice that it's impossible the trick here is if the shape that you get does not look like a cucumber cut in half so in this case our three-dimensional object looks kind of like a mountain with a den inside it and it does not look like a cucumber shape cut in half you have to use the cylindrical shell method and the shell method says that if we have a cylinder if we put the cylinder here and the base of that cylinder is on the x-axis then you integrate with respect to x if the base of the cylinder is on the y-axis so if the cylinder looks like this and the base of the cylinder is on the y-axis then we integrate with respect to y in this case it's on the x axis so the volume formula will simply be the integral from a to b of 2 times pi times the radius times the height d of x is just the thickness of the shell so it's basically the distance from here here that's going to be the thickness of the shell right there so what this formula is saying is from a to b so from a right here and b this is a and that's b we're gonna take the volume of each shell so we're going to take the volume of this shell right there and then we're going to take the volume of this shell moving from here to here right so the volume of this shell and then we're going to take the volume of this shell right here and then as we move towards b we're gonna take the volume of this shell right there and then we're gonna take the volume of this shell so i think you understand what i mean and then finally we're just going to take the volume of the final shell so this shell right there and the integral sign right here means that we add all of those volumes of our shells together and that's going to give us the volume of our three dimensional object now let's find the volume of this shape using the formula so step number one is to find a and b so a is gonna be in the middle so a is at the center and therefore a is equal to zero and b is right here so b is equal to two so the integral goes from zero to two the second step is to find the radius and the height so looking at the shell we know that the radius is just x from here to here so the radius r of x is equal to x now how about the height f of x the height is just y so f of x is equal to y and we know that y is equal to this right here so that's going to be 2 times x to the power of 2 minus x to the power of 3. step number three and the last step is to evaluate this integral so since 2 pi is just a constant we can move it outside of the integral so we're going to get 2 times pi times the integral from 0 to 2 and x times this we're going to get 2 times x to the power of 3 minus x to the power of 4 dx and this is the same as 2 times pi times the anti-derivative of 2 times x to the power of 3 and if you find the anti-derivative you're going to get 1 over 2 times x to the power of four minus one over five times x to the power of five and the limit goes from two to zero if you substitute these limits in you're going to get 2 times pi times 1 over 2 and then when you put the 2 in here you're going to get 16 because 2 to the power of 4 is 16 minus 1 over 5 and when you put a 2 in you're gonna get 2 to the power of 5 which is just 32 and when you substitute the 0 everything else is going to be 0. so this is our substitution and we finished it and this is equal to 2 times pi times 1 over 2 times 16 which is 8 minus 32 over 5. now 8 is the same thing as 40 divided by 5 so this is 40 divided by 5 and then 40 over 5 minus 32 over 5 is the same thing as 8 over 5. so this is 8 over 5 and when you multiply these two together the answer will simply be 16 divided by 5 times pi and you can leave it the way it is or you can say unit to the power of three so there we go this is the volume of our three-dimensional object using the cylindrical shell you