This resource contains ten linked problems which are intended to lead to students developing a rule for integrating functions of the form y=axn where n not equal to -1.
Problem 1 begins by exploring the area enclosed under a straight line y=2x+3 from x=0 to x=a for varying values of a, leading to generalisation. This involves finding the area of trapezia.
Problem 2 extends to generalising the area under the line y=mx+c between x=a and x=b.
Problem 3 asks students to explore what happens when one of the limits is a negative value.
Problem 4 requires students to generalise the area under a straight line for different values of m and c, and covering x=0 to x=a as a varies.
The next section explores the area under a curve. Beginning by finding upper and lower bounds for the area under the curve y=x2 between x=1 and x=2 by finding areas of rectangles.
Problem 5 asks students to find upper and lower bounds for the area between x=1 and x=3, using 3 and 6 rectangles.
Problem 6 extends the task to finding upper and lower bounds for the area under the curve y =1/x between x=1 and x=4.
The following section explores the idea of splitting the area into a number of trapezoidal strips with problem 7 asking students to find the area under the curve y=x2 between limits using a number of strips.
Problems 8 and 9 require students to generalise using n strips whilst problem 10 explores the area under the curve y=axn.
The teacher notes provided could be adapted to lesson plans and presentations. They include detailed explanations as well as solutions to each of the problems. There are a range of problems that progress in difficult to include examples to challenge more able students.