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anonymous
 5 years ago
The volume of the solid that results when y=6/7x^2 and y=13/7x^2 is rotated around the xaxis.
anonymous
 5 years ago
The volume of the solid that results when y=6/7x^2 and y=13/7x^2 is rotated around the xaxis.

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0If you look at the attached image, you'll see you need to find the volume that's generated by rotating the area between the two parabolas about the xaxis. If the top curve is y_2 and the bottom one, y_2, the elemental volume will be given by\[\delta V \approx \pi (y_2^2y_1^2)\delta x\](it's just area of circle generated by top curve multiplied by some 'thickness' minus area of bottom circle multiplied by some 'thickness', to give you a net element of volume). So you'd have,\[y=\pi \int\limits_{1}^{1}y_2^2y_1^2dx=\pi \int\limits_{1}^{1}(\frac{13}{7}x^2)^2(\frac{6}{7}x^2)^2dx\]\[=\pi \int\limits_{1}^{1} \frac{13x^4}{49}\frac{26}{7}x^2+\frac{169}{49}dx\]\[=\frac{3328}{735}\pi\]

dumbcow
 5 years ago
Best ResponseYou've already chosen the best response.1First graph the functions to get the bounds is it bounded by yaxis? we also need to know where they meet 6/7 x^2 = 13/7 x^2 solve for x > x = + 1 this means we will integrate from 0 to 1 when this graph is rotated about xaxis it creates a donut like solid with circular crosssections Area of each crosssection is piR1^2  piR2^2 where R1 is outer radius and measures distance from xaxis to yvalue of our upper function. upper function = 13/7 x^2 lower function = 6/7x^2 V = integral from 0 to 1 of (piR1^2  piR2^2) dx Factor out the pi R1^2 = (13/7 x^2)^2 = 169/49  26/7x^2 +x^4 R2^2 = (6/7x^2)^2 = 36/49x^4 R1^2  R2^2 = 13/49x^4  26/7x^2+169/49 Factor out 1/49 V =pi/49* integral from 0to1 of 13x^4  (26*7)x^2+169 dx V = pi/49*[13/5  ((26*7)/3) +169] = pi/49 *(110.933) = 2.26pi This assumes graph bounded by yaxis If not multiply volume by 2

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0The limits 1 and 1 are those xvalues where the two functions intersect, since this defines the upper and lower bounds of your interval.

dumbcow
 5 years ago
Best ResponseYou've already chosen the best response.1oh i didnt see you were answering...your explanation looks better haha

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0It's okay :) I like your name...
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