Question 4.7.2: Estimating an Integral from a Table of Function Values Esti...
Estimating an Integral from a Table of Function Values
Estimate \int_0^1 f(x) d x, where we have values of the unknown function f(x) as given in the table shown in the margin.
Approaching the problem graphically, we have five data points. (See Figure 4.27a.) How can we estimate the area under the curve from five points?
Conceptually, we have two tasks. First, we need a reasonable way to connect the given points. Second, we need to compute the area of the resulting region. The most obvious way to connect the dots is with straight-line segments as in Figure 4.27b.
Notice that the region bounded by the graph and the x-axis on the interval [0, 1] consists of four trapezoids. (See Figure 4.27c.)
It’s an easy exercise to show that the area of a trapezoid with sides h1 and h2 and base b is given by \left(\frac{h_1+h_2}{2}\right) b. (Think of this as the average of the areas of the rectangle whose height is the value of the function at the left endpoint and the rectangle whose height is the value of the function at the right endpoint.)
The total area of the four trapezoids is then
\frac{f(0)+f(0.25)}{2} 0.25+\frac{f(0.25)+f(0.5)}{2} 0.25+\frac{f(0.5)+f(0.75)}{2} 0.25 +\frac{f(0.75)+f(1)}{2} 0.25= [f(0)+2 f(0.25)+2 f(0.5)+2 f(0.75)+f(1)] \frac{0.25}{2}=1.125.
