Consider a function that is continuous on and sampled at equally spaced points with…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Consider a function

h

that is continuous on

[a,b]

and sampled at

n

equally spaced points

x_0=a,\,x_1,\,\dots,\,x_n=b

with common spacing

\Delta x=\dfrac{b-a}{n}

. Denote the left-endpoint sum by

L_n=\displaystyle\sum_{k=0}^{n-1} h(x_k)\,\Delta x

and the right-endpoint sum by

R_n=\displaystyle\sum_{k=1}^{n} h(x_k)\,\Delta x

. For each

k

, form the quadrilateral bounded by the straight-line segment joining

(x_k,h(x_k))

(x_{k+1},h(x_{k+1}))

, the

x

-axis, and the vertical lines

x=x_k

and

x=x_{k+1}

. The sum of the areas of all such quadrilaterals is denoted

T_n

. (a) Show algebraically that

T_n=\dfrac{L_n+R_n}{2}

for every positive integer

n

. Provide a clear derivation starting from the definition of

T_n

. (b) Let

h(x)=x^3

[0,2]

. Using

n=4

equally spaced subintervals, compute

L_n

R_n

, and

T_n

. (c) Based on the concavity of

h

, determine whether

T_n

is an overestimate or an underestimate of

\displaystyle\int_{0}^{2}x^3\,dx

. Justify completely.