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 x-axis to form a solid of revolution whose cross sections are washers.

d. Write an integral for the volume of the solid.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says consider the region T bounded by Y is equal to 3 plus the square root of X, X equal to 16, and Y equal to 3. If T is revolved about the X-axis, what is the integral or the volume of the solid formed using washers? So we're going to begin by just creating a quick sketch to determine our region team. So we're going to be looking in the first quadrant, so we'll go ahead and draw our X and Y axis for the first quadrant. And we're going to draw each of the three curves that we were given in the problem. So the first curve that we're given is Y is equal to 3 plus the square root of X. And so we'll begin at the point of 0.3 and just draw a sketch of the square root of X curve. And we started at 0.3 because when X is equal to 0, Y is equal to 3. That's gonna be a curve. Y is equal to 3 plus the square root of X. The next curve that we need to draw is X equal to 16, so that is going to be a vertical line. At X equal to 16. And the final curve that we need to draw is Y is equal to 3, so that's gonna be a horizontal line. At y equal to 3 and note that the curves of Y is equal to 3 plus the square root of X and Y equal to 3 intersect on the Y axis at the point of 0.3. And so the region bounded by these three curves is going to be our region T that we need to revolve about the X axis. Now, since we're revolving about the x-axis, that means that our washers that are formed by that revolution are going to have both an inner and outer radius. And so we need to determine what those are. And so we're going to begin by looking at our outer radius, which we'll call our sub out, and this is going to be equal to the curve that bounds our region T from above. And so if we look at the curve that is bounding our region T from above, we see that that is going to be Y is equal to 3 plus the square root of X. So that means that our out is going to be equal to 3, plus the square root of X. We'll now have to determine our inner radius, which we'll call our sub N. And this is going to be equal to the curve that is bending our region from below. And if we look at our sketch here, we see that the curve that is bending our region from below is Y is equal to 3, so that means that our N is just going to be equal to 3. And so now we can set up our integral for the volume of the solid formed using washers. So we'll call that volume V and recall that this is going to be equal to the integral, and here we have to look at our limits of integration, and our region T is between X equal to 0 and X equal to 16, since the curves Y equal to 3 plus the square x and y equal to 3 intersect at X equal to 0, and then on the right side we have the curve X equal to 16 bounding the region T. So that means that our integral for V is going to go from 0 to 16, and here we call your formula for finding the volume of the solid formed using washers, and that's going to be pi, multiplied by the quantity of R square minus Rn squad in quantity DX. So substituting in our expressions for R out and RN, we see that our volume is going to be equal to. The integral from 0 to 16 of pi. Multiplying the quantity of the quantity of 3 plus the square root of X in quantity squared minus 3 squad. In quantity DX. So this here is the integral that we are looking for for this problem. So that is the answer that we were looking for. So I hope that this explanation has been useful, and I'll see you 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 x-axis to form a solid of revolution whose cross sections are washers.

d. Write an integral for the volume of the solid.

Key Concepts

Solid of Revolution

Washer Method

Definite Integral

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