Evaluating a definite integral using the power rule antiderivative

Using the power rule for antiderivatives, $\int x^{1/3}\,dx = \frac{x^{4/3}}{4/3} + C = \frac{3}{4}x^{4/3} + C$. Since we only need one antiderivative for FTC Part 2, take $F(x) = \frac{3}{4}x^{4/3}$.

By FTC Part 2, $\int_1^8 x^{1/3}\,dx = F(8) - F(1) = \frac{3}{4}(8)^{4/3} - \frac{3}{4}(1)^{4/3}$. Since $8^{1/3}=2$, we get $8^{4/3} = 2^4 = 16$, so the expression becomes $\frac{3}{4}(16) - \frac{3}{4}(1) = 12 - \frac{3}{4} = \frac{45}{4}$.

The correct answer is lower limit. By FTC Part 2, the integral equals the antiderivative evaluated at the upper limit minus the antiderivative evaluated at the lower limit. In the worked example, this meant computing F of eight minus F of one, so the value at the lower limit, one, was the quantity being subtracted.