3.3 Evaluate the Riemann sum for 𝑓(𝑥) = 2 + 6𝑥, 2 ≤ 𝑥 ≤ 14, with six sub intervals taking the sample point to be left end points. Hence evaluate the following integral
14
∫ (2 + 6𝑥) 𝑑𝑥.
2
with 6 subintervals, the left endpoints are 3,4,6,8,10,12
The interval width is 2, so just add up
2(f(2)+f(4)+...+f(12))
To evaluate the Riemann sum for the function f(x) = 2 + 6x with left endpoints, we will divide the interval [2, 14] into six equal subintervals.
Step 1: Find the width of each subinterval.
The width of each subinterval is calculated by taking the difference between the upper and lower limits of integration and dividing it by the number of subintervals. In this case, the width will be (14 - 2) / 6 = 2.
Step 2: Determine the left endpoints of each subinterval.
Starting from the lower limit of integration, add the width of each subinterval to find the left endpoints. In this case, the left endpoints will be 2, 4, 6, 8, 10, and 12.
Step 3: Evaluate f(x) at each left endpoint.
Plug in the values of x into the function f(x) = 2 + 6x to find the corresponding y-values. In this case, the y-values will be f(2), f(4), f(6), f(8), f(10), and f(12).
- f(2) = 2 + 6(2) = 14
- f(4) = 2 + 6(4) = 26
- f(6) = 2 + 6(6) = 38
- f(8) = 2 + 6(8) = 50
- f(10) = 2 + 6(10) = 62
- f(12) = 2 + 6(12) = 74
Step 4: Calculate the Riemann sum.
Multiply each y-value by the width of the subinterval and sum them up. In this case, the Riemann sum will be:
Riemann sum = 2(14) + 2(26) + 2(38) + 2(50) + 2(62) + 2(74) = 56 + 52 + 76 + 100 + 124 + 148 = 556
Step 5: Evaluate the integral.
Finally, to evaluate the definite integral:
∫ (2 + 6x) dx from 2 to 14,
we can use the Riemann sum as an approximation. So the integral is approximately equal to the Riemann sum: 556.
Note: As the number of subintervals increases, the Riemann sum becomes a better approximation of the actual integral value. In this case, we used six subintervals.