23. Consider the function / (x) = x] + 1 to answer the following. a) Find the upper sum estimate for the area under the curve (to 4 decimal places) and sketch the area you are finding. [ Hint: Does this use left- or right-hand endpoints?] b) Find the lower sum estimate for the area under the curve (to 4 decimal places) and sketch the area you are finding. [Hint: Does this use left- or right-hand endpoints?] c) Find an estimate the area under the curve (to 4 decimal places) using the Midpoint Rule and sketch the area you are finding. Here are some helpful formulas for the following question: K = n(n + 1) 42 - "(at + 1)(2n + 1) 2 6 1 3 =1 24. Consider the function / (x) = 4x - 3x" over [0.1] to answer the following. a) Evaluate the definite integral of / using the definition with Riemann sums. b) Evaluate the same integral using the Fundamental Theoremn of Calculus. Is your answer the same as part (a)? If not, should it be? 25. Consider the function / (x) = x' + 4x over [0,3] to answer the following. a) Find the total area under the curve / (x)- b) Find the average value av(f ). 26. Find the derivative g'(x) for each of the following. al g(x) = [ vi+ sect at bj g(x) = [, cos'e de c) g(x) = = dt d) g(x) = ff Int at 27. Evaluate the following indefinite or definite integrals. (Use FTC where necessary.) a) (sin 2x - 25 ) dx b) ( + 1 -Vi) dx 2423 dx mye (2 + tan' 8) do [Hint: 1 + tan' 0 = sec] 0]