Question

The function y=f(x) is presented by the graph in the textbook. To estimate the area under the graph on the interval [-4, 4], we use left-rectangle approximation with four rectangles and obtain that the area is approximately

Answer 1

To estimate the area under the graph using the left-rectangle approximation, we divide the interval [-4, 4] into four equal subintervals of width 2.

First, we find the height of each rectangle by evaluating the function at the left endpoint of each subinterval. Let's assume the left endpoint of each subinterval is given by x = a.

For the first rectangle, with left endpoint x = -4, we have f(-4) as the height.
For the second rectangle, with left endpoint x = -2, we have f(-2) as the height.
For the third rectangle, with left endpoint x = 0, we have f(0) as the height.
For the fourth rectangle, with left endpoint x = 2, we have f(2) as the height.

After evaluating the function at each left endpoint, we calculate the area of each rectangle by multiplying the height by the width (2).

The area of the first rectangle: width * height = 2 * f(-4)
The area of the second rectangle: width * height = 2 * f(-2)
The area of the third rectangle: width * height = 2 * f(0)
The area of the fourth rectangle: width * height = 2 * f(2)

Finally, we sum up the areas of all the rectangles to estimate the total area under the graph on the interval [-4, 4].

Approximate area ≈ (2 * f(-4)) + (2 * f(-2)) + (2 * f(0)) + (2 * f(2))