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JAMB Mathematics 2010 Paper

Question : 44

Total: 49

Evaluate

\int\limits_{0}^{2}(x^3 + x^2)

dx.

Solution:

Concept:

For a definite integral, first integrate term by term, then evaluate the result from the lower limit to the upper limit.

Explanation:

Integrate the expression:

\int (x^3 + x^2)\,dx = \frac{x^4}{4} + \frac{x^3}{3} + C

Evaluate from

0

2

\int_{0}^{2} (x^3 + x^2)\,dx = \left[\frac{x^4}{4} + \frac{x^3}{3}\right]_{0}^{2}

Substitute the upper limit

x = 2

and the lower limit

x = 0

= \left(\frac{2^4}{4} + \frac{2^3}{3}\right) - \left(\frac{0^4}{4} + \frac{0^3}{3}\right)

= \left(\frac{16}{4} + \frac{8}{3}\right) - 0

= 4 + \frac{8}{3} = \frac{12}{3} + \frac{8}{3} = \frac{20}{3}

Convert to a mixed number:

\frac{20}{3} = 6\frac{2}{3}

Answer:

6\frac{2}{3}

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