Use the region R that is bounded by the graphs of y=1+√x,x=4, ...

Question

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.

Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.

c. What is the area A(y) of a cross section of the solid at a point y in [1, 3]?

Accepted Answer

Hello. In this video, we are going to be considering the region T bounded by Y is equal to 3 plus the square root of X, X is equal to 16, and Y is equal to 3. When the region T is revolved about the y axis, what is the area A of Y of cross section at height Y for Y on 3 to 7? So, this problem is asking us to find a function for the area of A in terms of Y. So, in order to start this problem, let's just go ahead and graph the information that is given to us. So, by drawing the axis, the first thing we're going to do is we're going to graph our first curve, which is going to be Y is equal to 3 plus the square root of X. Now, this is the standard square root function that is shifted up 3 units. So this is going to be the shape of that curve. Next, we need to draw X is equal to 16, and Y is equal to 3. Now, Y equal to 3 is a horizontal line. That passes through the value of 3 on the Y axis, and X is equal to 16 is a vertical line that passes through the value of 16 along the X axis. So, this is going to be the shape of the bounded region. So this is going to be a region capital T. Now, one more thing that this problem tells us is that this region is revolving about the Y axis. So our axis of rotation is going to be the Y axis. Now, one way that we can define the area is we can define the area using the washer method. See here, when we are looking at this area of rotation with respect to the shape of our axis, we have two radius for the axis. The first radius is going to be the distance from the Y axis to the vertical line, X is equal to 16. Then we have a smaller radius that's defined as the distance. From the y axis to the curve 3 + square root of X, so we have two different radi radiuses with a small gap if we were to revolve, so we need to use the washer method. That means that our area function Ay is going to be defined as pi multiplied by this bigger radius squared minus the smaller radius squared. Now, again, the bigger radius is going to be the distance from the Y axis to X's equal to 16. So that means that our capital R in this case is going to be defined as 16. Now, little R is the small distance from the Y axis to the smaller curve, which is going to be 3 + square root of X. But since we are going to integrate with respect to X, we want this to be in terms of Y. Since we are revolving about the Y axis. So, if we take that curve, Y is equal to 3 plus the square root of X. And so for X, the first thing we're going to do is subtract 3 to both sides of the equation, leaving us with square root of X is equal to Y minus 3, and squaring both sides is going to give us Y minus 3 squared. So, our smaller radius in this case is going to be Y minus 3 squared. And so now if we plug this into the area function, we have the area function in terms of Y to equal to pi multiplied by the quantity 16 2 minus Y minus 3 quantity squared, raised to the power of 2. And by further simplifying, our area function is going to be defined as pi multiplied by the quantity, 256 minus Y 3 to the fourth power, and this is going to be the solution to this problem. And the overall solution to this problem is going to be C. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises. Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers. c. What is the area A(y) of a cross section of the solid at a point y in [1, 3]?

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Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.

Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.

c. What is the area A(y) of a cross section of the solid at a point y in [1, 3]?