Question

Use sigma notation to represent the area under the curve y=1/x​ on the interval [1, 4] using left hand approximation and 6 subintervals.

Answer 1

To approximate the area under the curve y=1/x​ on the interval [1, 4] using left hand approximation and 6 subintervals, we follow next steps:

1. We divide the interval [1; 4] into 6 subintervals of equal length, ∆x = (4-1)/6 = 0.5. This divides the interval [1; 4] into 6 subintervals: [1, 1.5], [1.5, 2], [2, 2.5], [2.5, 3], [3, 3.5], [3.5, 4] each with length 0.5.

Above each subinterval draw a rectangle with height equal to the height of the function at the left endpoint of the subinterval:

1. We use the sum of the areas of the approximating rectangles to approximate the area under the curve. We get:

$A \approx L_6=\sum_{n=1}^{6} f(x_{i-1})∆x.$

$A \approx f(x_0)∆x+f(x_1)∆x+f(x_2)∆x+f(x_3)∆x+f(x_4)∆x+f(x_5)∆x=$

$=∆x(f(x_0)+f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5))=$

$=\frac{1}{2}(1+\frac{2}{3}+\frac{1}{2}+\frac{2}{5}+\frac{1}{3}+\frac{2}{7})=\frac{1}{2}(3\frac{13}{70})=\frac{1}{2}(\frac{223}{70})=1\frac{83}{140} \approx 1.59$