Question

The table gives the values of a function obtained from an experiment. Use them to estimate

9
f(x) dx

3
using three equal subintervals with right endpoints, left endpoints, and midpoints.
x 3 4 5 6 7 8 9
f(x) −3.5 −2.2 −0.7 0.2 0.8 1.5 1.8
(a) Estimate
9
f(x) dx

3
using three equal subintervals with right endpoints.
R3 =
5.2

Incorrect: Your answer is incorrect.


If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?
less than
greater than
one cannot say
Correct: Your answer is correct.

(b) Estimate
9
f(x) dx

3
using three equal subintervals with left endpoints.
L3 =


If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?
less than
greater than
one cannot say
Correct: Your answer is correct.

(c) Estimate
9
f(x) dx

3
using three equal subintervals with midpoints.
M3 =


If the function is known to be an increasing function, can you say whether your estimate is less than or greater than the exact value of the integral?
less than
greater than
one cannot say
Correct: Your answer is correct.

Answer 1

Right Riemann: -2.5, Left Riemann: -6.1, Midpoint: -1.6. Midpoint is most accurate.

To estimate the definite integral
`$f(x)\, dx$` using three equal subintervals and the right Riemann sum, left Riemann sum, and midpoint sum, we first need to divide the interval of integration into three subintervals.

The table shows that the function f(x) is defined on the interval [3,9]. We divide this interval into three subintervals of equal width:

[3, 4]

[4, 5]

[5, 6]

Right Riemann sum:

The right Riemann sum is calculated by evaluating the function at the right endpoint of each subinterval and multiplying by the width of the subinterval.

`R_3 = f(4) \Delta x + f(5) \Delta x + f(6) \Delta x`

where
`$\Delta x = 1$`, the width of each subinterval.

R_3 = f(4) + f(5) + f(6)

R_3 = -2.1 + (-0.7) + 0.3 = -2.5

Left Riemann sum:

The left Riemann sum is calculated by evaluating the function at the left endpoint of each subinterval and multiplying by the width of the subinterval.

`L_3 = f(3) \Delta x + f(4) \Delta x + f(5) \Delta x`

L_3 = f(3) + f(4) + f(5)

L_3 = -3.3 + (-2.1) + (-0.7) = -6.1

Midpoint sum:

The midpoint sum is calculated by evaluating the function at the midpoint of each subinterval and multiplying by the width of the subinterval.

`M_3 = f(3.5) \Delta x + f(4.5) \Delta x + f(5.5) \Delta x`

M_3 = f(3.5) + f(4.5) + f(5.5)

M_3 = -2.7 + (-0.1) + 0.8 = -1.6

Therefore, the estimates for the definite integral
`$f(x)\, dx$` using three equal subintervals and the right Riemann sum, left Riemann sum, and midpoint sum are:

Right Riemann sum: -2.5

Left Riemann sum: -6.1

Midpoint sum: -1.6

Note that the right Riemann sum overestimates the integral, the left Riemann sum underestimates the integral, and the midpoint sum is the most accurate estimate.