R is the region bounded by the functions fx 1 x2 and the gx...

R is the region bounded by the functions f(x) = 1 + x^2 and the g(x) = √x + 1 and the lines x=0 and x=4. Represent the volume when R is rotated around the line x=4. Volume = ∫_0^4 dx

Transcript

00:01 I hope i'm reading this problem correctly, that it's x squared divided by four, because x to the two over four power doesn't make a whole lot of sense.

00:09 So that's g of x as well as f of x being x.

00:17 And so i guess what i'm interested in is in this graph, if i were to graph f of x, it's just that diagonal line, and then, you know, go ahead and label it f, and then g of x is a parabola, but it is a flattened out one.

00:39 So there's g, and i'll just label it over here.

00:42 So if i'm interested in where they intersect each other, i need to set the equations equal to each other.

00:49 At this point, i would multiply four to the right side.

00:53 And if you're familiar with factoring, subtract four x to the other side, and then you can have the greatest common factor of x in front, and then you can see that the points of intersection will be at zero right here, and then also at four over here.

01:11 And then the last thing that we need to understand is that you're revolving around the horizontal line y equals negative one.

01:18 So that's down here.

01:20 Now what i always have students do is notice that you have this empty space, and when you have this space and you start to revolve things around, let's do it in green, so you're revolving this way, that you have a washer.

01:37 I don't know if you can see that washer.

01:38 I'm not very artistic either, so that's not helping.

01:42 So it says write the definite integral...

Question

R is the region bounded by the functions f(x) = 1 + x^2 and the g(x) = √x + 1 and the lines x=0 and x=4. Represent the volume when R is rotated around the line x=4. Volume = ∫_0^4 dx

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