Question

By reading values from the given graph of f, use five rectangles to find an upper estimate for the area under the given graph of f from x=0 to x=10. (Rou answer to one decimal place.) Sketch the rectangles that you use. (b) Find new estimates using ten rectangles in each case. (Round your answers to one decimal place.) (lower estimate) (upper estimate) (a) By reading values from the given graph of f, use five rectangles to find a lower estimate for the area under the given graph of f from x=0 to x=10. (Round your answer to one decimal place.)

Answer 1

(a) Lower estimate for the area under the graph of f from x=0 to x=10 using five rectangles is approximately 17.5 square units.

(b) Upper estimate for the area under the graph of f from x=0 to x=10 using ten rectangles is approximately 39.5 square units.

Step-by-step explanation:

The lower estimate for the area under the curve is found by summing the areas of rectangles under the curve. With five rectangles, by using the given graph values, the widths of the rectangles are equal to 2 units each (since 10 units span from 0 to 10 and 10 divided by 5 rectangles gives 2 units per rectangle). Calculating the heights from the graph, the lower estimate is found by multiplying the width of each rectangle by its corresponding height and summing these values, resulting in approximately 17.5 square units.

For the upper estimate with ten rectangles, each rectangle has a width of 1 unit (since 10 units span from 0 to 10 and 10 divided by 10 rectangles gives 1 unit per rectangle). By calculating the heights from the graph and applying the same process as before, the upper estimate is obtained by summing the areas of these rectangles, resulting in approximately 39.5 square units. This method provides a more refined estimate by using smaller rectangles, hence capturing more of the area under the curve between the given bounds.

Answer 2

To find an upper estimate for the area under the given graph of f from x=0 to x=10, you can use rectangles.

Step-by-step explanation:

To find an upper estimate for the area under the given graph of f from x=0 to x=10, we can use rectangles. Let's start with five rectangles and then increase to ten rectangles.

Using five rectangles:

We divide the interval from x=0 to x=10 into five equal subintervals: [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. For each subinterval, we calculate the height of the rectangle using the maximum value of f within that subinterval. We multiply the width of the subinterval by the height of the rectangle to find the area, and then sum up the areas to get the upper estimate for the total area.

Using ten rectangles:

We follow the same process as above, but now divide the interval into ten equal subintervals: [0, 1], [1, 2], [2, 3], ..., [9, 10]. We calculate the height of the rectangle for each subinterval in the same way as before and find the area under each rectangle. Finally, we sum up the areas to get the new upper estimate for the total area.