Final answer:
To estimate the area under the graph of f(x) = 3x^3 + 4, we used three methods: right endpoints, left endpoints, and midpoints with 6 and then 12 rectangles. For each method, we calculated the height at specified points and multiplied by the width Δx to estimate the area.
Step-by-step explanation:
To estimate the area under the graph of f(x) = 3x3 + 4 from x = -1 to x = 5, we will calculate the area using approximating rectangles (as in the Riemann sum method) following three different approaches: using right endpoints, left endpoints, and midpoints. The width of each rectangle (Δx) will be determined by dividing the total interval length by the number of rectangles (6 or 12 in this case).
Right Endpoints with 6 Rectangles: The interval from x = -1 to x = 5 has a length of 6, so Δx will be 6/6=1. We calculate the right-endpoint height of each rectangle by evaluating f(x) at x = 0, 1, 2, 3, 4, 5 and use these values to approximate the total area by summing up the areas of each rectangle.
Right Endpoints with 12 Rectangles: Here we use Δx = 6/12=0.5, and the corresponding right-endpoint x-values would be x = -0.5, 0, 0.5, 1, ..., 4.5, 5. We calculate the height of each rectangle and sum up the areas to improve the estimate of the total area.
Using Left Endpoints
Left Endpoints with 6 Rectangles: Using the same approach as above but with the left-endpoint of each rectangle (Δx = 1), the x-values would be x = -1, 0, 1, 2, 3, 4. The heights are evaluated at these points to estimate the total area.
Left Endpoints with 12 Rectangles: Δx is 0.5, with the left-endpoint x-values being x = -1, -0.5, 0, ..., 3.5, 4. We calculate the heights and areas of each rectangle and sum them to get a better estimation.
Using Midpoints
Midpoints with 6 Rectangles: We use midpoints of intervals x = -0.5, 0.5, 1.5, ..., 4.5 to calculate the heights of the rectangles (Δx = 1) and thus estimating the area.
Midpoints with 12 Rectangles: The midpoints would be x = -0.75, -0.25, 0.25, ..., 4.25, 4.75 and Δx = 0.5. By calculating the height of each rectangle and their areas, we sum up the values for a more accurate area estimation under the curve.