5.6. NUMERICAL INTEGRATION
321
(c) Use the Fundamental Theorem of Calculus to compute the exact value of I =
3
0
x 2 dx.
(d) We define the error in an approximation of a definite integral to be the difference
between the integral’s exact value and the approximation’s value. What is the
error that results from using L 3 ? From R 3 ? From M 3 ?
(e) In what follows in this section, we will learn a new approach to estimating the
value of a definite integral known as the Trapezoid Rule. The basic idea is to use
trapezoids, rather than rectangles, to estimate the area under a curve. What is the
formula for the area of a trapezoid with bases of length b 1 and b 2 and height h?
(f) Working by hand, estimate the area under f (x) = x 2 on [0, 3] using three subintervals and three corresponding trapezoids. What is the error in this approximation?
How does it compare to the errors you calculated in (d)?
⊲⊳
The Trapezoid Rule
Throughout our work to date with developing and estimating definite integrals, we have
used the simplest possible quadrilaterals (that is, rectangles) to subdivide regions with
complicated shapes. It is natural, however, to wonder if other familiar shapes might serve
us even better. In particular, our goal is to be able to accurately estimate
b
a
f (x) dx
without having to use extremely large values of n in Riemann sums.
To this end, we consider an alternative to L n , R n , and M n , know as the Trapezoid Rule.
The fundamental idea is simple: rather than using a rectangle to estimate the (signed) area
bounded by y = f (x) on a small interval, we use a trapezoid. For example, in Figure 5.15,
we estimate the area under the pictured curve using three subintervals and the trapezoids
that result from connecting the corresponding points on the curve with straight lines.
The biggest difference between the Trapezoid Rule and a left, right, or middle Riemann
sum is that on each subinterval, the Trapezoid Rule uses two function values, rather than
one, to estimate the (signed) area bounded by the curve. For instance, to compute D 1 ,
the area of the trapezoid generated by the curve y = f (x) in Figure 5.15 on [x 0 , x 1 ], we
observe that the left base of this trapezoid has length f (x 0 ), while the right base has length
f (x 1 ). In addition, the height of this trapezoid is x 1 − x 0 = △x =
b−a
3 . Since the area of a
trapezoid is the average of the bases times the height, we have
D 1 =
1
2
( f (x 0 ) + f (x 1 )) · △x.
Using similar computations for D 2 and D 3 , we find that T 3 , the trapezoidal approximation
321
(c) Use the Fundamental Theorem of Calculus to compute the exact value of I =
3
0
x 2 dx.
(d) We define the error in an approximation of a definite integral to be the difference
between the integral’s exact value and the approximation’s value. What is the
error that results from using L 3 ? From R 3 ? From M 3 ?
(e) In what follows in this section, we will learn a new approach to estimating the
value of a definite integral known as the Trapezoid Rule. The basic idea is to use
trapezoids, rather than rectangles, to estimate the area under a curve. What is the
formula for the area of a trapezoid with bases of length b 1 and b 2 and height h?
(f) Working by hand, estimate the area under f (x) = x 2 on [0, 3] using three subintervals and three corresponding trapezoids. What is the error in this approximation?
How does it compare to the errors you calculated in (d)?
⊲⊳
The Trapezoid Rule
Throughout our work to date with developing and estimating definite integrals, we have
used the simplest possible quadrilaterals (that is, rectangles) to subdivide regions with
complicated shapes. It is natural, however, to wonder if other familiar shapes might serve
us even better. In particular, our goal is to be able to accurately estimate
b
a
f (x) dx
without having to use extremely large values of n in Riemann sums.
To this end, we consider an alternative to L n , R n , and M n , know as the Trapezoid Rule.
The fundamental idea is simple: rather than using a rectangle to estimate the (signed) area
bounded by y = f (x) on a small interval, we use a trapezoid. For example, in Figure 5.15,
we estimate the area under the pictured curve using three subintervals and the trapezoids
that result from connecting the corresponding points on the curve with straight lines.
The biggest difference between the Trapezoid Rule and a left, right, or middle Riemann
sum is that on each subinterval, the Trapezoid Rule uses two function values, rather than
one, to estimate the (signed) area bounded by the curve. For instance, to compute D 1 ,
the area of the trapezoid generated by the curve y = f (x) in Figure 5.15 on [x 0 , x 1 ], we
observe that the left base of this trapezoid has length f (x 0 ), while the right base has length
f (x 1 ). In addition, the height of this trapezoid is x 1 − x 0 = △x =
b−a
3 . Since the area of a
trapezoid is the average of the bases times the height, we have
D 1 =
1
2
( f (x 0 ) + f (x 1 )) · △x.
Using similar computations for D 2 and D 3 , we find that T 3 , the trapezoidal approximation