The diagram shows a sketch of the curve y = f(x), where

The region R, shown in the diagram, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = 4. The table below shows the corresponding values of x and y with the y values given to 4 decimal places as appropriate.

a. Complete the table with the missing value of y.

b. Use the trapezium rule, with all the values of y in the table, to obtain an estimate for the area of R, giving your answer to 3 decimal places.

c. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of R.

d. Show that the exact area of R can be written in the form a/b + c/d ln e , where a, b, c, d and e are integers.

e. Find the percentage error in the answer in part b.

Transcribed Image Text:

f(x) = x² ln x − x + 2, x > 0.