(a) Verify that the average of the left and right end-point approximations as given in Table 7.7.1 gives Formula (2) for the trapezoidal approximation. (b) Suppose that f is a continuous nonnegative function on the interval [a, b] and partition [a, b] with equally spaced points, a = x 0 < x 1 < ... < x n = b . Find the area of the trapezoid under the line segment joining points ( x k , f ( x k ) ) and ( x k + 1 , f ( x k + 1 ) ) and above the interval x k , x x k + 1 . Show that the right side of Formula (2) is the sum of these trapezoidal areas (Figure 7.7.1).
(a) Verify that the average of the left and right end-point approximations as given in Table 7.7.1 gives Formula (2) for the trapezoidal approximation. (b) Suppose that f is a continuous nonnegative function on the interval [a, b] and partition [a, b] with equally spaced points, a = x 0 < x 1 < ... < x n = b . Find the area of the trapezoid under the line segment joining points ( x k , f ( x k ) ) and ( x k + 1 , f ( x k + 1 ) ) and above the interval x k , x x k + 1 . Show that the right side of Formula (2) is the sum of these trapezoidal areas (Figure 7.7.1).
(a) Verify that the average of the left and right end-point approximations as given in Table 7.7.1 gives Formula (2) for the trapezoidal approximation.
(b) Suppose that
f
is a continuous nonnegative function on the interval [a, b] and partition [a, b] with equally spaced points,
a
=
x
0
<
x
1
<
...
<
x
n
=
b
.
Find the area of the trapezoid under the line segment joining points
(
x
k
,
f
(
x
k
)
)
and
(
x
k
+
1
,
f
(
x
k
+
1
)
)
and above the interval
x
k
,
x
x
k
+
1
.
Show that the right side of Formula (2) is the sum of these trapezoidal areas (Figure 7.7.1).
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.