Indefinite integration

Given that

Hence find

The area, A, bounded by the curve with equation y 2^x, the x-axis, the y-axis and the line x 4 is approximated using eight rectangles of equal width as shown in the diagram below.

Show that the exact area of the largest rectangle is

The areas of these rectangles form a geometric sequence with common ratio

Find the exact value of the total area of the eight rectangles. Give your answer in the form k(1 2) where k is a rational number.

More accurate approximations for A can be found by increasing the number, n, of rectangles used.

Find the exact value of the limit of the approximations for A as n

The binomial expansion of (2 5x)⁴ is given by

4 2 3 4

Find the value of A and the value of B.

Show that

4 4 3

where C and D are constants to be found.

Hence, or otherwise, find

Find

Show that

for small values of x.

Hence, show that the area between the graph with equation

the positive x-axis and the line x 025 can be approximated by

where m and n are integers to be found.

Explain why

is not a suitable approximation for

Explain how

may be approximated by

for suitable values of a and b.