That equation should look very familiar to you - it's the positive radical of an equation of a circle. In other words, your graphs looks like the top half of a circle, or a semicircle.
I'm telling you this because for these problems, you should draw a diagram to help you out. For AP exams, you might even get extra points for doing so.
Draw your circle, which should have its center at (9,0) and a radius of 4. Draw an arrow curling around the y-axis to show that you're spinning it around that axis.
Since you're going around a vertical axis, and your equation is in terms of x, use the shell method, in which you use an infinite number of shells to find the volume of a rotated function (just as you use an infinite number of discs and washers in the disc and washer methods). Every teacher writes the equation a little differently, so don't be alarmed if what I write here looks different than what you normally use:
V = 2pi times the integral from a to b of p(x) times h(x) dx
p(x) is the distance from the axis (either x or y) and h(x) is the length of the shell (hence the shell method), generally the function being rotated. The values for a and b are the limits of integration, the starting and stopping points of the rotated shape.
This is why drawing a picture is helpful - so you can see what a, b, h, and p are. In your case, a and b would be where the endpoints of your semicircle are, which are x = 5 and x = 13. h(x) is just the equation y (the height, or y-value of the function), and since we're in terms of x, p(x) is x.
Plug in your numbers/variables, and integrate. There you go!
the area defined is going from y=0 to a million, yet for x it is going from x=0 to infinity. to discover the quantity while revolved around the y axis, you're including up a team of tiny cylinders: dV = (2 Pi x f(x) dx) So the indispensable would be: indispensable (x = 0 to infinity) (2Pi x (a million - ((e^x)-(e^-x)) / ((e^x)+ (e^-x)))) i don't understand an hassle-free thank you to take this indispensable, yet it is the respond... yet permit's attempt this: 2Pi * indispensable (0 - Inf) (x [(e^x + e^-x) - (e^x - e^-x)] / (e^x + e^-x)) = 2Pi * indispensable (0 - Inf) (x [2e^-x] / (e^x + e^-x)) = 2Pi * indispensable (0 - Inf) (x [2] / (e^2x + a million)) = 2Pi * indispensable (0 - Inf) (2x / (e^2x + a million)) Ed: even although the form is going out to infinity, it could have a finite volume. it quite is an argument of despite if it converges or not. because of the fact the indispensable sounds like x/e^2x, i'm making a wager that it will converge. the quantity pertaining to to the x-axis could be: indispensable(0 to infinity) of (Pi f(x)^2 dx) = indispensable(0 to infinity) of (Pi (a million^2 - (((e^x)-(e^-x)) / ((e^x)+ (e^-x))^2) dx)
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That equation should look very familiar to you - it's the positive radical of an equation of a circle. In other words, your graphs looks like the top half of a circle, or a semicircle.
I'm telling you this because for these problems, you should draw a diagram to help you out. For AP exams, you might even get extra points for doing so.
Draw your circle, which should have its center at (9,0) and a radius of 4. Draw an arrow curling around the y-axis to show that you're spinning it around that axis.
Since you're going around a vertical axis, and your equation is in terms of x, use the shell method, in which you use an infinite number of shells to find the volume of a rotated function (just as you use an infinite number of discs and washers in the disc and washer methods). Every teacher writes the equation a little differently, so don't be alarmed if what I write here looks different than what you normally use:
V = 2pi times the integral from a to b of p(x) times h(x) dx
p(x) is the distance from the axis (either x or y) and h(x) is the length of the shell (hence the shell method), generally the function being rotated. The values for a and b are the limits of integration, the starting and stopping points of the rotated shape.
This is why drawing a picture is helpful - so you can see what a, b, h, and p are. In your case, a and b would be where the endpoints of your semicircle are, which are x = 5 and x = 13. h(x) is just the equation y (the height, or y-value of the function), and since we're in terms of x, p(x) is x.
Plug in your numbers/variables, and integrate. There you go!
the area defined is going from y=0 to a million, yet for x it is going from x=0 to infinity. to discover the quantity while revolved around the y axis, you're including up a team of tiny cylinders: dV = (2 Pi x f(x) dx) So the indispensable would be: indispensable (x = 0 to infinity) (2Pi x (a million - ((e^x)-(e^-x)) / ((e^x)+ (e^-x)))) i don't understand an hassle-free thank you to take this indispensable, yet it is the respond... yet permit's attempt this: 2Pi * indispensable (0 - Inf) (x [(e^x + e^-x) - (e^x - e^-x)] / (e^x + e^-x)) = 2Pi * indispensable (0 - Inf) (x [2e^-x] / (e^x + e^-x)) = 2Pi * indispensable (0 - Inf) (x [2] / (e^2x + a million)) = 2Pi * indispensable (0 - Inf) (2x / (e^2x + a million)) Ed: even although the form is going out to infinity, it could have a finite volume. it quite is an argument of despite if it converges or not. because of the fact the indispensable sounds like x/e^2x, i'm making a wager that it will converge. the quantity pertaining to to the x-axis could be: indispensable(0 to infinity) of (Pi f(x)^2 dx) = indispensable(0 to infinity) of (Pi (a million^2 - (((e^x)-(e^-x)) / ((e^x)+ (e^-x))^2) dx)