Evaluate \(\int_{0}^{2} (8x - 4x^{2}) \mathrm {d} x\).
Question 1 Report
Evaluate \(\int_{0}^{2} (8x - 4x^{2}) \mathrm {d} x\).
Answer Details
To evaluate \(\int_{0}^{2} (8x - 4x^{2}) \mathrm {d} x\), we first need to find the antiderivative of the integrand with respect to x.
\[\int (8x - 4x^{2}) \mathrm {d} x = 4x^{2} - \frac{4}{3} x^{3} + C\]
where C is the constant of integration.
Using the limits of integration, we can evaluate the definite integral as follows:
\[\int_{0}^{2} (8x - 4x^{2}) \mathrm {d} x = \left[4x^{2} - \frac{4}{3} x^{3}\right]_{0}^{2}\]
\[= \left[4(2)^{2} - \frac{4}{3}(2)^{3}\right] - \left[4(0)^{2} - \frac{4}{3}(0)^{3}\right]\]
\[= \frac{16}{3}\]
Therefore, the answer is \(\frac{16}{3}\).
Option (c) is the correct answer.
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