t (seconds) 1 3 4 7
v(t) (meters/second) 20 32 42 52
The table shows the velocity (in meters per second) for an object over the interval [1.7]. Estimate \int^7_1v\left(t\right)dt∫17v(t)dt using 3 subintervals and the Left Hand Riemann Sum.
To estimate the integral ∫7└1 v(t) dt using 3 subintervals and the Left Hand Riemann Sum, we need to divide the interval [1, 7] into 3 equal subintervals.
First, let's find the width of each subinterval. We can calculate it by subtracting the lower limit from the upper limit and dividing by the number of subintervals:
Width of each subinterval = (7 - 1) / 3
= 6 / 3
= 2
Now we can construct the subintervals:
Subinterval 1: [1, 3]
Subinterval 2: [3, 5]
Subinterval 3: [5, 7]
Next, we need to find the left endpoint of each subinterval. In this case, the left endpoint is the starting point of each subinterval. Here are the left endpoints for each subinterval:
Left endpoint of Subinterval 1: 1
Left endpoint of Subinterval 2: 3
Left endpoint of Subinterval 3: 5
To estimate the integral using the Left Hand Riemann Sum, we find the velocity value at each left endpoint and multiply it by the width of each subinterval. Finally, we sum up these products.
Estimate of the integral ∫7└1 v(t) dt using the Left Hand Riemann Sum is:
(left endpoint of Subinterval 1, v(t) at 1) x (Width of each subinterval) +
(left endpoint of Subinterval 2, v(t) at 3) x (Width of each subinterval) +
(left endpoint of Subinterval 3, v(t) at 5) x (Width of each subinterval)
Plugging the values into the formula:
Estimated integral ≈ (1, 20) x 2 + (3, 32) x 2 + (5, 42) x 2
Calculating this expression:
Estimated integral ≈ (1)(20)(2) + (3)(32)(2) + (5)(42)(2)
= 40 + 192 + 210
= 442
Therefore, the estimate of the integral ∫7└1 v(t) dt using 3 subintervals and the Left Hand Riemann Sum is approximately equal to 442 meters.