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**HallsofIvy** Have you graphed the region to be rotated? Using the disk method, each "disk", at a value of y, will have radius equal to 4 minus the x value corresponding to that y. For y from 0 up to the y-value where the two curves cross, the radius will be $\displaystyle 4- (4- x- x^4)= x^4+ x$. From the y-value where the two curves cross, the radius will be 4- (10 ln(x)). The crucial question is "where do those two curves cross?". Can you solve $\displaystyle 10ln(x)= 4- x- x^2$?