Some values of the function ft et4 are given in the table b

Some values of the function f(t) = e-t4 are given in the table below. By setting up a Riemann sum using 7 subintervals, find an upper estimate to the definite integral Use your calculator where appropriate, and show all working.

Solution

Riemann sums require you to draw rectangles of width (delta)x, or the change in x. The height is dependent on what kind of Riemann sums you\'re drawing. In this problem, it says upper limits, so within the rectangle of width (delta)x, you would choose the higher function value. Then, you know the width and height, multiply and find the area. There are 7 subintervals, so there will be 7 rectangles you must calculate. Once you have all 7, you will add them all up.

Examples

The first subinterval has a width of 0.25 - 0, or 0.25. The height is f(0) = 1 because this rectangle is bounded by x = 0 and x = 0.25. Here f(0)>f(0.25). Multiply the width and height and you should get 0.25.

The second subinterval has a width of 0.5 - 0.25, or 0.25. The height is f(0.25) = 0.996 because this rectangle is bounded by x = 0.25 and x = 0.5. Here, f(0.25)>f(0.5). Multiply the width and height and you should get 0.249.

I won\'t do the rest, but this should help you with the rest of the rectangles. Note that while the 2 rectangles I used as examples had a width of 0.25, it\'s not always going to be such. The next 5 you\'ll see will have a width of 0.1.

REMEMBER: once you have all the rectangles, add them up to get your upper limit.