In this problem you will calculate ∫30x2+4x by using the formal definition of the definite integral: ∫(x)x=lim→[infinity][∑=1(x∗)Δx]. (a) The interval [0,3] is divided into equal subintervals of length Δx. What is Δx (in terms of )? Δx =
(b) The right-hand endpoint of the th subinterval is denoted x∗. What is x∗ (in terms of and )? x∗ =
(c) Using these choices for x∗ and Δx, the definition tells us that ∫30x2+4x=lim→[infinity][∑=1(x∗)Δx]. What is (x∗)Δx (in terms of and )? (x∗)Δx =
(d) Express ∑=1(x∗)Δx in closed form. (Your answer will be in terms of .) ∑=1(x∗)Δx =
(e) Finally, complete the problem by taking the limit as →[infinity] of the expression that you found in the previous part. ∫30x2+4x=lim→[infinity][∑=1(x∗)Δx] =