Find the volume, V, of the solid obtained by rotating the...

Find the volume, V, of the solid obtained by rotating the region bounded by the graphs of $x = y$, $x = \sqrt{y}$ about the line $x = -1$. 1. $V = \frac{7\pi}{15}$ 2. $V = \frac{31\pi}{30}$ 3. $V = \frac{3\pi}{5}$ 4. $V = \frac{3\pi}{4}$ 5. $V = \frac{\pi}{2}$ 6. $V = \frac{9\pi}{10}$ 7. $V = \frac{2\pi}{5}$ 8. $V = \frac{29\pi}{30}$

Transcript

00:01 So for this question, we're asked to find the volume of the solid obtained by rotating the region bound by the given curves around the y axis.

00:11 So we're given the curves x equals four times the square root of five y.

00:20 So the y is inside.

00:23 X equals zero and y equals three.

00:29 All right.

00:30 So let's go ahead and sketch that graph real quick to give ourselves an idea here.

00:36 So let's start the two easier ones here.

00:38 So x equals zero, x equals zero, that's simply the y axis.

00:43 And y equals three, that's a vertical line, or sorry, horizontal line through the y axis at three.

00:50 And let's transform this one, that's square both sides here of this one.

00:55 If i square both sides, i'm going to get x squared equals.

00:59 If i square the four, i get 16 times squared the square to five.

01:04 I just get five y.

01:06 Right.

01:07 So that's x squared equals.

01:13 80y which is y equals 1 over 80 x squared well that's a parabola right y equals something x squared that's a parabola so let's bring this out here and really i don't even need to know where they intersect right i can find out where they intersect by taking this this is the one intersects right so if i substitute a three in for here i can solve for x all right so we want to revolve around the y axis right so this is this integral is easily done in terms of y so i'm going to do something d y and i can see here that my y goes from zero to three from zero to three right and what i want to do is i want to add up these discs right these are just discs not washer because they go all the way to the axis and so remember the area is going to be uh pi r squared the area of each disk right it's going to be pi r squared so that's going to be pi times the radius well we can see here that the radius is just from zero to this function right here so right that the function is going to give me the the radius so this is going to be four times the square root of five y and remember the area is pi r squared well from here i can take pi to the outside and then i can apply that square.

02:51 So let's go ahead and do that.

02:56 So pi here, the integral from zero to three...

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