Answer:
7
Step-by-step explanation:
To approximate the area under the function f(x) = x³ - 1 between a = 1 and b = 3 using a left-hand sum, we divide the interval [1, 3] into two subintervals of equal width and then sum the areas of the rectangles formed by the function values at the left endpoints of these subintervals.
First, find the width of each subinterval:
[tex]\Delta x = \dfrac{b - a}{n} = \dfrac{3 - 1}{2} = 1[/tex]
The left endpoints of the subintervals are x = 1 and x = 2. So, the area of each rectangle is the product of the value of f(x) at the left endpoints and the width of each subinterval:
[tex]A = f(1) \cdot \Delta x + f(2) \cdot \Delta x[/tex]
[tex]A = (1^3-1)\cdot 1 + (2^3-1) \cdot 1[/tex]
[tex]A = (1-1)\cdot 1 + (8-1) \cdot 1[/tex]
[tex]A = (0)\cdot 1 + (7) \cdot 1[/tex]
[tex]A=0+7[/tex]
[tex]A=7[/tex]
So, the left-hand sum approximation of the area under the function f(x) = x³ - 1 between a = 1 and b = 3 using 2 intervals is 7.
Note that the left-hand sum provides a lower estimate of the area under the curve in this case. The exact area under the curve is 18.
