Volume with disc method: revolving around other axes
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Calculating integral disc around vertical line
In the last video, we had set up the definite integral to evaluate the volume of this upside-down gumdrop truffle-looking thing. So now in this video, we can actually evaluate the definite integral. So what we need to do is really just expand out this expression, this square root of y plus 1 plus 2. So let's do that. So this is going to be equal to pi times the definite integral from y is equal to negative 1 to y is equal to 3. If you expand this out, you get square root of y plus 1 squared, which is just going to be y plus 1. And then you're going to have 2 times the product of both of these terms. 2 times 2 times square root of y plus 1 is going to be plus 4 times the square root of y plus 1. And then you have 2 squared, so plus 4. So you have this whole thing times dy. We can simplify a little bit. You have a 1 plus a 4. We can add the 1 to the 4 and get a 5. And now we're ready to take the antiderivative. So this is going to be equal to pi times-- let's take the antiderivative of all of this business-- pi times-- and I'll color code it. The antiderivative of y is just y squared over 2. The antiderivative of 4 times the square root of y plus 1-- you just really have to think of it as 4 times y plus 1 to the 1/2 power. We could use u substitution explicitly, but you probably are pretty practiced in this and can do this in your head. You have y plus 1 raised to the 1/2 power. Derivative of y plus 1 is just 1, which is essentially out here. So if you did u substitution, you would say u is equal to y plus 1. But this antiderivative is going to be equal to-- well, if you increment this exponent, you get 3/2 multiplied by the reciprocal 2/3. 2/3 times 4 is 8/3. So it's plus 8/3 times y plus 1 to the 3/2. And you can verify. If you take the derivative here you will get this expression right over here. 3/2 times 8/3 is 4. Decremented, you have y plus 1 to the 1/2 power. And then finally, you have-- let's see. What color have I not used yet? Finally, you have this 5. The antiderivative of 5 is just 5y. And we are going to evaluate it at 3 and at negative 1, y equals 3 and y equals negative 1. So this is going to be equal to pi. So let's evaluate all this business at 3. So 3 squared over 2 is 9/2. 3 plus 1 is 4 to the 3/2. Well, that's-- so let's see. If square root of 4 is 2 to the third power is 8, 8 times 8/3 is 64/3, so plus 64/3. You have 5 times 3. Well, that's going to be 15-- plus 15. And from that, we're going to subtract all this business evaluated at negative 1. So you have negative 1 squared over 2. Well, that's just 1/2. Negative 1 plus 1 is 0 to the 3/2 power. That's going to be 0 times 8/3. This is all going to be 0, so we don't have to even write it. And then finally, you have negative 1 times 5. Well, that's just going to be negative 5. And we are in the home stretch. We really just have to do a little bit of arithmetic, add some hairy fractions right over here. So let's do it. So this whole thing is going to simplify to pi times-- and it looks like-- let's see. Our least common multiple of all of these denominators is going to be 6. So let's put everything over a denominator of 6. So 9/2 is the same thing as 27/6. 64/3 is the same thing as 128/6. 15 is the same thing as 90/6. 1/2 is the same thing as 3/6. So you would distribute the negative sign. So this is negative 3/6. And negative times negative is positive. 5 is the same thing as 30/6, so plus 30/6. And so this is going to give us-- our denominator is going to be over 6. We're going to multiply something times pi. We have this pi over here. And then we just have to figure out what the numerator is. So let's see if I can do this in my head. So 27 plus 128 is going to be-- let me see. That's going to be 140, 155? Is that right? 155? Let's see, if we get to 48 plus another 7-- yeah, 155. Plus 90 gets us to 245. Is that right? Yeah. Plus 90 gets us to 245. Minus 3, you subtract 3 from that, you get to 242. And then you add 30 to that, you get to 272. So we're left with 272 pi/6. But then we can-- let's see. 272 and 6 are both divisible by 2. So this is equal to-- let's see. 272 divided by 2 is going to be 136 pi over-- and if you divide this denominator right over here by 2-- over 3. Is that right? Yeah. 136 pi/3. And 136 is not divisible by 3. So we have it as simplified as we can. This right over here is the volume of our little upside-down gumdrop-looking thing.
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