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Gennadij [26K]
4 years ago
14

Use a trapezoidal sum with the four sub-intervals indicated by the data in the table to estimate the definite integral from 0 to

8 of R of t, dt. Using correct units, explain the meaning of your answer in terms of water flow. Give 3 decimal places in your answer.
t(hours) 0 | 2 | 3 | 7 | 8
R(t) (gallons per hour) 1.95 | 2.5 | 2.8 | 4 | 4.26

Mathematics
1 answer:
Elis [28]4 years ago
3 0
   \displaystyle\int_0^8 R(t) \, dt

   \begin{array}{l | c c c c c}
t \text{ (hours)} & 0 & 2 & 3 & 7 & 8 \\
R(t) \text{ (gallons per hour)} & 1.95 & 2.5 & 2.8 & 4 & 4.26
\end{array}

The area of a trapezoid is given by \frac{1}{2}(a+b)(h). If we take the height to be the change in x, then 1/2\big[R(t_i) + R(t_{i-1})\big](\Delta t) is the area of one trapezoid for one subinterval with borders t_i and t_{i-1}. Therefore, we approximate as follows, using the data to dictate our trapezoid bases and heights:

   \displaystyle\int_0^8 R(t) \, dt \\ \\
\approx \tfrac{1}{2} (R(2) + R(0))(2-0) + \tfrac{1}{2} (R(3) + R(2))(3-2) \\ \\
{}\qquad + \tfrac{1}{2} (R(7) + R(3))(7-3) + \tfrac{1}{2} (R(8) + R(7))(8-7)  \\ \\
&= \tfrac{1}{2}(2.5 + 1.95)(2) + \tfrac{1}{2} (2.8+2.5)(1) \\ \\
{}\qquad + \tfrac{1}{2} (4 + 2.8 )( 5) + \tfrac{1}{2} (4.26+4 )(1) \\ \\
&=  24.83 \text{ gallons}

From t = 0 hours to t = 8 hours, the total amount of water that flows out is approximately 24.83 gallons.

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