Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)
(a) The signed area between the graph of f(x) = cos x and the x-axis on [−π, π].
(b) The average value of f(x) = cos x on [0, 2π].
(c) The area of the region between the graphs of f(x) =
Part (a) 0
Part (b) 0
Part (c) 12.56
The function is,.
The objective is to find the signed area on without calculating.
Consider the following graph of the function:
From the graph it is seen that two areas are negative while two areas are positive of equal areas.
Therefore, the signed area is 0.
The function is,
The objective is to find the average value on without calculating.
From the above graph it is seen that both the positive area bis equal to the negative area.
Therefore, the average value is 0.
For each function f and interval [a, b] in Exercises 27–33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.
, n = 3 with
a) Trapezoid sim b) Upper sum
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