You can solve for the area under curves with integrals. I don't know what level math you are in, but if the word intergrals rings a bell, then you should know what to do. Since you are given the limits of the area, you would use a definite intergral.
I don't know how to show the intergral notation online, so I'll just tell you that the intergral of the cube root of x is (3/4)*x^(4/3) and then to solve for the given limits you would plug the upper limit (8) into the intergral, then subtract the intergral with the lower limit (0) plugged in:
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