1 point) \'y=g(x) Consider the blue vertical line shown above (click on graph fo

Question

1 point) 'y=g(x) Consider the blue vertical line shown above (click on graph for better view) connecting the graphs yg(x) - sin(x) and y =f(x) = cos(2x). Referring to this blue line, match the statements below about rotating this line with the corresponding statements about the result obtained. i 1. The result of rotating the line about the x-axis is 2. The result of rotating the line about the y-axis is iii 3. The result of rotating the line about the line y = 1 is 4. The result of rotating the line about the line x =-2 is 5. The result of rotating the line about the line x is EE 6. The result of rotating the line about the line y--2 is 7. The result of rotating the line about the line y = is 8. The result of rotating the line about the line y =- is A, a cylinder of radius -x and height cos(2x)-sin(x) B. an annulus with inner radius 2sin(x) and outer radius 2 +cos(2x) C. an annulus with inner radius + sin(x) and outer radius + cos(2x) D. an annulus with inner radius -cos(2x) and outer radius -sin(x) E. an annulus with inner radius 1 - cos(2x) and outer radius 1- sin(x) F an annulus with inner radius sin(x) and outer radius cos(2x) G. a cylinder of radius x and height cos(2x) -sin(x) H. a cylinder of radius x + 2 and height cos(2x) sinx)

Explanation / Answer

It is to be noted that the line if rotated about x axis or some line parallel to x axis would result in an annulus

(vertical rotation) and rotation about y axis or some line parallel to y axis would result in a cylinder (horizontal rotation).

1. about x axis, the bottom most point of the line would trace out the outer circumference of the inner circle and the top most point would trace out the outer circumference of the outer circle. Hence, the inner radius would be y=g(x) and outer radius would be y=f(x) (Answer: F)

2. about y axis, the cylinder traced out would have a radius equal to the distance of the line from y axis. i.e. x and the height of the cylinder would be length of the line i.e. f(x) - g(x). (Answer: G)

3. about y=1, an annulus will be traced with inner radius = 1-f(x) (y=1 for f(x) where x=0, & as x increases f(x) decreases in value) and outer radius = 1-g(x) (y=g(x) being the lowest point of the line). (Answer: E)

4. about x = -2 ( a line parallel to y axis) , a cylinder would be traced out with radius equal to the distance between the line x=-2 and the current line i.e. x+2 and height would be the same length of the line cos(2x) - sin(x). (Answer: H)

5. about x = pi, again a cylinder would be traced out with radius equal to pi - x (as pi lies further on the x -axis) and height would be the same (length of the line). (Answer: A)

6. about y = -2, (a line parallel to x axis), an annulus would be traced out with inner radius equal to the distance between the line y=-2 and the lowermost point of the current line i.e. g(x) + 2 and outer radius equal to the distance between the line y=-2 and the uppermost point of the current line i.e. f(x) + 2. (Answer: B)

7. about y = pi ( a point that will lie way above both the curves), an annulus would be traced out with inner radius equal to pi - f(x) and outer radius equal to pi - g(x). (Answer: D)

8. about y = - pi, an annulus would be traced out again with inner radius equal to pi + g(x) (since g(x) will be closer to y = -pi line this time) and outer radius equal to pi + f(x). (Answer: C)

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