Question

Can someone make sure that my answer is right?

A.

Width = (b – a)/n
Width = (24 – 0)/3
Width = 8

Trapezoid rule:
8/2(f(0) + 2f(8) + 2f(16) f(24)
4(0.5 + 3.4 + 3 + 0.5)
4(7.4)
29.6

This definite integral should physically represent the total water usage for Seattle over this 24 hour period. Trapezoids approximate the total water usage as 29.6 million gallons/hr.

B.

0 < t < 8 is a “concave up” part of the interval. Therefore, the trapezoidal sum should overestimate the area.

C.

I had trouble writing out the integral as text, so my answer to Part C is attached as a second image.

D.

r’(12) = (r(14) – r(12))/(14 – 12)
r’(12) = (1.4 – 1.3)/2
r’(12) = 0.05

Answer 1

Comments on your answer:

A) The math is correct, but the units are not. When you multiply gallons per hour by hours, you get gallons. The result should be expressed as 29.6 Mgal (per 24-hour period).

B) The first and second parts of the curve are concave with respect to your estimating trapezoid, so your estimate can be high in those time periods. The third part of the curve is convex relative to your trapezoid, so you under-estimate that part. Overall, it looks like your approximation is an over-estimate.

C) Yes, 1.233 Mgal/hour (don't forget the units)

D) The curve has a change of slope at or near t=12. The slope at t<12 is negative, while at t>12 it is positive. If this were an absolute value function, we would say the slope is undefined at that point. If you don't simply want to declare the slope undefined, you can use an estimate from 10 to 14, which will give a slightly negative result: (1.4 -1.5)/(14 -10) = -0.025.