Question

Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval. (Round your answers to four decimal places.)

g(x) = 4 sin x, [0, ], 6 rectangles

Answer 1

The area under the curve g(x) = 4 sin x can be approximated using six rectangles based on both left and right endpoints over the interval [0, π], and the results are rounded to four decimal places.

Step-by-step explanation:

Approximating Area Using Left and Right Endpoints

The question pertains to estimating the area under the curve of the function g(x) = 4 sin x using rectangles. With six rectangles over the interval [0, π], we must use both left and right endpoints to approximate the areas.

To use the left endpoints, we divide the interval into 6 equal parts. Each rectangle's width is π/6, and the height is determined by the value of the function at the left end of each subinterval. The approximate area is the sum of the areas of these rectangles.

The right endpoints method is similar but uses the value of the function at the right end of each subinterval to determine each rectangle's height. The sum of the areas of these rectangles gives us a second approximation.

Finally, we round both approximations to four decimal places as requested.