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Equate areas using horizontal and vertical strips

For the curve y=x3y=x^3, the region between the curve, x=1x=1, x=qx=q and the x-axis has the same area as the region between the curve, y=1y=1, y=8y=8 and the y-axis, where q>1q>1. Find qq exactly as a1/ba^{1/b} for integers a,ba,b.

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Curve y=x cubed with two shaded areas: below the curve between x=1 and x=q, and to the left of the curve between y=1 and y=8. Dashed boundaries identify x=1,x=q,y=1,y=8; q is to the right of x=2.
Curve y=x cubed with two shaded areas: below the curve between x=1 and x=q, and to the left of the curve between y=1 and y=8. Dashed boundaries identify x=1,x=q,y=1,y=8; q is to the right of x=2.
Curve y=x cubed with two shaded areas: below the curve between x=1 and x=q, and to the left of the curve between y=1 and y=8. Dashed boundaries identify x=1,x=q,y=1,y=8; q is to the right of x=2.

[2016/MJC/I/2(b)]

CJC · Discussion · Question 6

Horizontal strips use x as a function of y.

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