Question

Estimate the area under the graph of f(x)=3x²−18x+28 over the interval [0,8] using eight approximating rectangles and right endpoints. Rn= Repeat the approximation using left endpoints. Ln= Report answers accurate to 4 places. Remember not to round too early in your calculations. Estimate the area under the graph of f(x)=9−x2 over the interval [−3,2] using ten approximating rectangles and right endpoints. Rn= Repeat the approximation using left endpoints. Ln= Report answers accurate to 4 places.

Answer 1

To estimate the area under a graph using right and left endpoints, divide the interval into equal subintervals and evaluate the function at the respective endpoint. Multiply the function value by the width of the subinterval and sum up these areas to get the estimate.

Step-by-step explanation:

To estimate the area under the graph of f(x) = 3x²−18x+28 over the interval [0,8] using eight right endpoints, we can divide the interval into eight equal subintervals. The width of each subinterval is (8-0)/8 = 1. We evaluate the function at the right endpoint of each subinterval and multiply the function value by the width of the subinterval. Adding up these areas gives us the estimate for the total area.

Similarly, to estimate the area using eight left endpoints, we use the left endpoint of each subinterval to evaluate the function.

For the given function f(x) = 9−x² over the interval [−3,2], we can follow the same process to estimate the area using ten right endpoints and ten left endpoints.