Use the definition to find an expression for the area under the curve y = x3 from 0 to 1 as a limit. lim n→∞ n i = 1 r (b) the following formula for the sum of the cubes of the first n integers is proved in appendix e. use it to evaluate the limit in part (a). 13 + 23 + 33 + + n3 = n(n + 1) 2 2

The summand (R?) is missing, but we can always come up with another one.

Divide the interval [0, 1] into subintervals of equal length :

Let's consider a left-endpoint sum, so that we take values of where is given by the sequence

with . Then the definite integral is equal to the Riemann sum

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b) k’(6, 4); l’(3, 3)

step-by-step explanation:

a 270° clockwise rotation maps every point (x, y) to (y, -x). (it negates the x-coordinate and then switches it and the y-coordinate.)

this means the endpoints of the side congruent to kl, k' and l', would be mapped as:

k'(-4, 6)→(6, 4)

l'(-3, 3)→(3, 3)