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please answer these questions showing all steps
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Math 273
Assignment 11
Due Date: December ¡++¿
Assignment
1. Compute each of the following definite integrals by applying the Fundamental Theorem of
Calculus Part Two.
Z 5
(x − 2)(5 − x) dx
(a)
2
Z π
sin2 (x) dx
(b)
−π
R 11
2. Suppose that f is continuous on the interval [4, 11] and that 4 f (x) dx = 35. Explain why
there must be a number c in the interval [4, 11] so that f (c) = 5.
R x2
3. Let g(x) = 0 f (t) dt, where f is the function whose graph is displayed below. Compute
g 0 (3).
4. Determine whether the following statements are true or false; justify your answer.
(a) If the function F (x) is an antiderivative of a function f (x), then
Z b
f (x) dx = F (b) − F (a).
a
(b) If a function f (x) is continuous on [a, b], then the area enclosed by the graph of y = f (x),
y = 0, x = a and x = b equals (numerically)
Z b
f (x) dx.
a
(c) If u = g(x) is a differentiable function whose range is an interval I and f is continuous
on I, then
Z
Z
0
f g(x) g (x) dx = f (u) du.
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