Trapezium Rule

Edexcel IAL WMA12/01/P2/Oct 2022/Q6 (The Trapezium Rule, Integration)

The curve C1 has equation y = f(x). A table of values of x and y for y = f(x) is shown below, with the y values rounded to 4 decimal places where appropriate.

• Use the trapezium rule with all the values of y in the table to find an approximation for​​ 

∫02fx dx

giving your answer to 3 decimal places.​​ 

(3)

The region R, shown shaded in Figure 1, is bounded by​​ 

• the curve C1​​ 

• the curve C2​​ with equation​​ y = 2 –14 x2​​ 

• the line with equation x = 2​​ 

• the y‑axis​​ 

The region R forms part of the design for a logo shown in Figure 2.​​ 

The design consists of the shaded region R inside a rectangle of width 2 and height 3​​ 

Using calculus and the answer to part (a),​​ 

• calculate an estimate for the percentage of the logo which is shaded.​​ 

(4)

SOLUTION​​ 

a- Using Trapezium Rule to estimate the value of​​ ∫02fxdx.

(The number of trapezium strips are 4 as we know that the number of strip is always 1 less than the total number of values of x and y given.)

h=b-an=2-04=24=12=0.5

A=0.52 3+1.92+22.6833+2.4+2.1466

∫02fxdx  ≈4.84495

=4.845 

b- ​​ 

(To find the shaded region of the logo design, subtract the area under​​ C2​​ from the area under​​ C1. As far as area under​​ C1​​ is concerned, we have found it in part (a) through trapezium rule. Whereas, the area under​​ C2​​ can be found by integrating the equation of​​ C2​​ within the limits 0 and 2. Once, we have the area of the shaded region, we will estimate the percentage of logo which is shaded.)

Area of R=∫02fxdx-∫022-14x2 dx

=4.845-2x-112x302 

=4.845-4-812

Area of R=1.5116

Now, to calculate the​​ percentage of logo which is shaded.​​ 

% of logo Shaded=Area RArea of Rectangular Logo Design×100

To find the​​ Area of Rectangular Logo Design​​ simply find the area of rectangle of dimensions 2×3. So the area will be 6 unit square.

% of logo Shaded=1.51166  x 100=25.1936

% of logo Shaded=25.2%