1M1 Section 3 video 1: follow-up quiz

When using the trapezium rule, we approximate the graph using ...

• straight lines.

• parabolae.

• exponential functions.

• trigonometric functions.

When using Simpson's rule, we approximate the graph using ...

• parabolae.

• straight lines.

• exponential functions.

• trigonometric functions.

What is the area of a trapezium with parallel sides that are 2cm apart and have lengths 3cm and 4cm?

(a)  

Answer explanation

The area of a trapezium is the base times the average of the heights.

The general formula to apply the trapezium rule for an integral from $a$ to $b$ is $A\approx\frac{h}{2}\left(y_0+2y_1+...+2y_{N-1}+y_N\right)\$ . What is the relationship between $h$ and $N$ in this formula?

•  $h=\frac{b-a}{N}$  

•  $h=\frac{a-b}{N}$  

•  $h=\left(b-a\right)N^{ }$  

•  $h=a+bN$  

Answer explanation

Remember $h$ is the width of the subintervals and $N$ is the number of subintervals!

In both the trapezium and Simpson's rule, we use the notation $y_j=f\left(x_j\right)$ . If we have divided the interval from $a$ to $b$ into $N$ subintervals, which of the following is the correct formula for $x_j$ ?

•  $x_j=a+j\ \frac{b-a}{N}$  

•  $x_j=b+j\ \frac{b-a}{N}$  

•  $x_j=a+jN$  

•  $x_j=b+jN$