Water leaks from a tank at the rate of r(t) gallons per hour. The rate decreased as time passed, and values of the rate at two-hour time intervals are shown in the table below. The total amount of water that leaked out is evaluated by a Riemann sum. Find the upper estimate (left end-points of each rectangle) for the total amount of water that leaked out by using five rectangles.

Give your answer with one decimal place.

Question

Water leaks from a tank at the rate of r(t) gallons per hour. The rate decreased as time passed, and values of the rate at two-hour time intervals are shown in the table below. The total amount of water that leaked out is evaluated by a Riemann sum. Find the upper estimate (left end-points of each rectangle) for the total amount of water that leaked out by using five rectangles. 

Give your answer with one decimal place.

anonymous · Answer

https://gyazo.com/945c50dd03c9c13bfe58ff4bd2c269d8

agl202 · Answer

Left Sum = 2 * (r(0) + r(2) + r(4) + r(6) + r(8)) 
= 2 * (10.7 + 8.6 + 6.6 + 5.2 + 5.0) 
= 72.2

anonymous · Answer

@Agl202 could you write of out the Rieman Sum/ Sigma Notation?

anonymous · Answer

@jim_thompson5910 @zepdrix  would any of you be willing to give me a hand? I'd really like to know the procedure not just the answer.

jim_thompson5910 · Answer

does this pic help?

anonymous · Answer

Yes but is there calculus way of solving this? Like writing out the Riemann sum?

jim_thompson5910 · Answer

@Agl202 wrote out the expanded form of the Riemann sum

the step just before it would be something like this

$$\Large \sum_{k=0}^{n-1}r(x_k)*\Delta x$$

jim_thompson5910 · Answer

Since delta x is constant, we can pull it out


$$\Large \sum_{k=0}^{n-1}r(x_k)*\Delta x=\Delta x*\sum_{k=0}^{n-1}r(x_k)$$

anonymous · Answer

How do I find Xk?

jim_thompson5910 · Answer

there are 5 rectangles, so n = 5 and n-1 = 5-1 = 4

the x_k is the general term for x_1, x_2, x_3, x_4

jim_thompson5910 · Answer

k starts at 0 and goes to 4 (k is a whole number)

jim_thompson5910 · Answer

each x_k value is found by looking at the first row of the table

notice how @Agl202 plugged in x = 0, x = 2, etc etc

jim_thompson5910 · Answer

btw the answer isn't 72.2 but it's pretty close

@Agl202 made a typo

anonymous · Answer

I don't think I got it then. If he plugged in those values he wouldn't get those numbers.

anonymous · Answer

Wait are you just adding what's in the table?

anonymous · Answer

@jim_thompson5910

jim_thompson5910 · Answer

yeah he used the wrong numbers from the table

jim_thompson5910 · Answer

but the idea is still the same

anonymous · Answer

So basically add then multiply by two?

jim_thompson5910 · Answer

yeah because delta x (or delta t) is equal to 2

jim_thompson5910 · Answer

keep in mind that the last value in the table is NOT used (look at how the rectangles are set up in my drawing to see why)

anonymous · Answer

I got 70 for the final answer. @jim_thompson5910

jim_thompson5910 · Answer

8.7+7.6+6.8+6.2+5.7 = 35

2*35 = 70

same here