2.
We find the average value of the function, f(x), on the interval [a, b] using the formula f(c) =
f(x) dx. Here we are looking for the average value of f(x) = on the interval [0,
16]. Using the formula, we need to find dx. We get =
= = = .
3. 1
We find the average value of the function, f(x), on the interval [a, b] using the formula f(c) =
f(x) dx. Here we are looking for the average value of f(x) = 2|x| on the interval [−1,
1]. Using the formula, we need to find (2|x|) dx. Recall that the absolute value
function must be rewritten as a piecewise function:|x| = . Thus, we need to split the
integral into two separate integrals in order to evaluate it: =
x dx. We get = =
= 1.
- sin^2 x
We find the derivative of an integral using the Second Fundamental Theorem of Calculus:
f(t) dt = f(x). We get sin^2 t dt = sin^2 x.
- 27 x^2 − 9x
We find the derivative of an integral using the Second Fundamental Theorem of Calculus:
