Two methods The region R in the first quadrant bounded by the ...

Question

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:

a. Apply the disk method and integrate with respect to y.

a. Apply the disk method and integrate with respect to y.

Accepted Answer

Welcome back, everyone. In this problem, let T be the region in the first quadrant bounded by the parabola Y equals 1 minus 4th of X squared and the coordinate axis. When T is revolved above the y axis, what is the volume of the resulting solid using the DC method? A says it's pi cubic units. B 2 cubic units. C. 4 cubic units, and D 8 pi cubic units. Now, if we're going to figure out the volume of the solid using the disco method, there are a few things that we need. First, we need the bones of our region, OK? Now, we're told that it's bounded by the parameter and the coordinate axis, and the equation for the coordinate axes are X equals 0 and Y equals 0. Now, when X equals 0, OK. Then Y equals 1 minus 02 divided by 4 or 1. So Y runs from Y equals 0 to Y equals 1, and that's the bone that we're interested in because we are rotating, revolving it about the y axis, OK. So thus Y equals 0 and Y equals 1. Are the bones of our region, that is the bones of the. Next, what do we need? Well, since we're revolving it about the y axis, using the DC method we'll be integrating with respect to Y, which means we'll need to express our equation where X is the subject in terms of Y. So given our equation, Y equals 1 minus 4th of X2. Then that means that 1/4 of X squared must be equal to 1 minus Y, which means then that X2 equals 4 multiplied by 1 minus Y. And if that's the case, OK, let's make some space here. Then this must mean that X equals the square root of 4 multiplied by 1 minus Y or 2 multiplied by root 1 minus Y. Now that we have X in terms of Y, we can set up the disk method integral for revolution above the y axis. In other words, now we know the volume is going to be equal to the integral between Y equals 0 and Y equals 1 of pi multiplied by X, which is 2 multiplied by the square root of 1 minus Y squared, all with respect toy. Now, if we think about it, we already know that 2 multiplied by 1 minus Y squared is going to be equal to 4 multiplied by 1 minus Y. And if we go ahead and rewrite this, then we can write this as 4 pi multiplied by the integral uh between 0 and 1 of 1 minus Y with respect to Y. No, we can integrate 1 minus Y with respect to Y. Which means we're going to end up with 4 pi multiplied by y minus 2 of y squared between the bones of 0 and 1. Now we can substitute our bones. For our upper bound, that's going to be 1 minus 1 squared divided by 2 and for our lower bound if we substitute 0, both of those terms will become zero. And now if we expand that 4 pi multiplied, well, let's simplify it 1st. 1 minus 2 will be equal to 1/2 and 4 multiplied by 2 equals to a pi. Thus, the volume of the solid is 2 cubic units. Thanks a lot if we look back at our answer choices, my apologies. If we look back at our answer choices, that tells us then that B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:a. Apply the disk method and integrate with respect to y.

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Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:

a. Apply the disk method and integrate with respect to y.