Maths with David

    9231. FP2. Integration. Past Exam Questions

    9231/21/M/J/25q2

    [4 marks]

    (b) Find the exact value of I2. [3 marks]

    9231/21/M/J/25q4

    The diagram shows the curve with equation y = \frac{1}{ \sqrt{x} } e^{ \sqrt{x}} for x \geq 1 , together with a set of n-1 rectangles of unit width.

    (a) By considering the sum of the areas of these rectangles, show that \sum_{r=1}^n \frac{1}{ \sqrt{r}} e^{ \sqrt{r} } < ( 2 + \frac{1}{ \sqrt{n}} ) e^{ \sqrt{n} } - 2e [5 marks]

    (b) Use a similar method to find, in terms of n, a lower bound for \sum_{r=1}^n \frac{1}{ \sqrt{r} } e^{ \sqrt{r}} [4 marks].