77. The region in the first quadrant bounded by the coordinate...

Question

77. The region in the first quadrant bounded by the coordinate axes, the line y=3, and the curve x=2/√(y+1) is revolved about the y-axis to generate a solid. Find the volume of the solid.

Accepted Answer

Welcome back everyone. The region in the first quadrant bounded by the x axis, the horizontal line y equals 2, and the curve x equals 4 divided by y plus 1, is revolved about the y axis. Compute the volume of the solid generated a 4 pi divided by 3, B 8 pi divided by 3, C 16 pi divided by 3. And the 32 pi divided by 3. For this problem, we're going to use the disc method. Specifically, volume can be found by taking pi and multiplying by the integral from A to B of R Y. Squad Dy Where R of Y is the radius of the disk. The vertical bounds of the region. are going to be from 0 to 2, right? So we can show that a is equal to 0 and b is equal to 2. Notice that the region is bounded by the x axis and the horizontal line y equals 2. That's how we get a equals 0 and B equals 2. Now R of y is going to be our function 4 divided by. Y + 1. And we can now set up the integral volume equals the integral. From 0 to 2. We're going to write pi in front and now let's square our function which is going to give us 16, that's 4 squared, divided by y plus 1 square followed by d y. Now we have our integral set up. And we can integrate, so that's pi integral from 0 to 2. We can factor out 16 because it's a constant. And now we will rewrite 1 divided by y + 1 squad as y + 1 raised to the power of -2. Which is going to allow us to apply the power rule. So we get 16 pi. Integrating using the power rule. We will get Y + 1 to the power of -2 + 1, so that's -1 divided by -1, right? The slope of the linear term is one, so according to the reverse chain rule, we don't have to divide by anything. Well, technically we're dividing by one, but there's no necessity to include it. And now we want to calculate the result from 0 to 2. So let's go ahead and simplify first. That's going to be -16 divided by. Y + 1 from 0 to 2. Applying the limits of integration we get. 16. Divided by Y + 1, so that's 2 + 1, which is 3. 16 pi Divided by Y + 1 adds 0 is going to be 0 + 1, which is 1. Simplifying, we get 16 pi minus 16 pi divided by 3. And using the common denominator, that would be 48 5 minus 16 pi divided by 3. Which is 32 pi divided by 3. So the correct answer for this problem corresponds to the answer of choice the 32 pi divided by 3 cubic units. Thank you for watching.

77. The region in the first quadrant bounded by the coordinate axes, the line y=3, and the curve x=2/√(y+1) is revolved about the y-axis to generate a solid. Find the volume of the solid.

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