Exploring area linked to graphs

This activity, from Shirley Fall, contains ten linked problems which are intended to lead to students developing a rule for integrating functions of the form y=ax n 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=x 2 between limits using a number of strips. Problem 8 and 9 require students to generalise using n strips whilst problem 10 explores the area under the curve y=ax n. 

The teacher notes provide solutions to each of the problems.