Unformatted text preview: 7. Use the Fundamental Theorem of Calculus to show that
I cos(x) dx = 1
(without using your calculator).
8. Explain how you can tell, just by looking at the graph ofy = cos(x), that I: cos(x)dx = 0. 9. Verify that
j"cos(x)dx = 0
0 by using the FTC (without using your calculator). 10. Based on your work in the past few problems, explain you know that there is
some number, x1, between 7r/2 and it, such that j cos(x)dx = 0.686.
0 (Hint: You don't need to do any calculations. Just reflect on your results from the
previous few problems.)
11. Are there are two different numbers, x1 and x2, each between 7r/2 and 7r such that
I“ cos(x)dx = 0.686 and I“ cos(x)dx = 0.686?
Explain your answer.
12. Draw the graph of a function,ﬁ such that jjﬂxmxzjfﬁxwx. Indefinite Integrals The term “indeﬁnite integral” is often used to denote the set of all antiderivatives of
a function,f, on some interval (a,b). The reason for the use of this term is that
integration is closely related to antidifferentiation (by the FTC). We use the notation [(me ﬂx) dx to refer to the indefinite integral of the functionfon the interval (a,b) — meaning the set
of all functions that are antiderivatives offon the interval (a,b). The Mean Value
Theorem can be used to prove that if a function, F, is an antiderivative of the function f
on a certain interval ((1,1)), then any antiderivative, G, of f on (a,b) must be of the form
G(x) = F(x) + C where C can be any constant. Thus, for example, since we know that
the function F(x) = %x2 is an antiderivative of the function ﬂx) : x on the interval
(—oo,oo), then we know that any antiderivative, G, of f must be of the form G(x) = '72? + C (where C can be any constant). We describe the indefinite integral off as I xdx = -1—x2 + C.
(wow) 2 In most cases, it is not necessary to include the interval ((1,1)) in the notation for the 14 ...
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- Fall '08