With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of n, using looping. (On a TI use the Is> command or a For-End For loop, on a Casio use Is z, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for n = 10, 30, and 50. Then guess the value of the exact area.

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7. The region under y = sin x from 0 to 7 8. The region under y= 1/x* from 1 to 2