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# Limits : Evaluating Integrals, Anti-Differentiation and Area Between Curves

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(a) Consider the attached limit of summed terms
(i) Explain why each of the sums in the attached expression gives an over-estimate of the area beneath the curve {see attachment}
(ii) Evaluate this limiting sum, using the expression {see attachment}
(iii) Check your answer in (ii), by using the fundamental theorem of calculus to evaluate the corresponding definite integral usuing anti-differentiation

(b) Without evaluating the integral, find F'(x) where {see attachment}

(c) Write down, and evaluate, an integral expression for the area of the region enclosed by the two curves {see attachment} for values of x between x = 1 and x = 3. Sketch the region, and check your answer matches an approximation to the area based in your sketch.