(B) If , then u^2 = x + 1, and 2u du = dx. When you substitute for
the limits, you get . Since u ≠ 0 on its interval of integration,
you may divide numerator and denominator by it.
(D)
(D) On [0,6] with n = 3, Δx = 2. Heights of rectangles at x = 1, 3, and 5
are 5, 9, and 5, respectively; M(3) = (5 + 9 + 5)(2).
(D)
(E) For L(2) use the circumscribed rectangles: ; for R(2) use
the inscribed rectangles: .
(D) On [0,1] f(x) = cos x is decreasing, so R < L. Furthermore, f is
concave downward, so T < A.
(D) On the interval [−1,3] the area under the graph of y = |x| is the sum of
the areas of two triangles: .
(E) Note that the graph y = |x + 1| is the graph of y = |x| translated one
unit to the left. The area under the graph y = |x + 1| on the interval [−3,2]
