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4.3. THE DEFINITE INTEGRAL
(d) Use appropriate technology to compute Riemann sums to estimate the object’s
total distance travelled on [0, 4]. Work to ensure that your estimate is accurate
to two decimal places, and explain how you know this to be the case.
(e) What is the object’s average velocity on [0, 4], accurate to two decimal places?
3. Consider the graphs of two functions f and g that are provided in Figure 4.30. Each
piece of f and g is either part of a straight line or part of a circle.
1
2
3
4
-2
-1
1
2
y = f (x)
1
2
3
4
-2
-1
1
2
y = g(x)
Figure 4.30: Two functions f and g.
(a) Determine the exact value of
1
0
[ f (x) + g(x)] dx.
(b) Determine the exact value of
4
1
[2 f (x) − 3g(x)] dx.
(c) Find the exact average value of h(x) = g(x) − f (x) on [0, 4].
(d) For what constant c does the following equation hold?
4
0
c dx =
4
0
[ f (x) + g(x)] dx
4. Let f (x) = 3 − x 2 and g(x) = 2x 2 .
(a) On the interval [−1, 1], sketch a labeled graph of y = f (x) and write a definite
integral whose value is the exact area bounded by y = f (x) on [−1, 1].
(b) On the interval [−1, 1], sketch a labeled graph of y = g(x) and write a definite
integral whose value is the exact area bounded by y = g(x) on [−1, 1].
(c) Write an expression involving a difference of definite integrals whose value is
the exact area that lies between y = f (x) and y = g(x) on [−1, 1].
(d) Explain why your expression in (c) has the same value as the single integral
1
−1
[ f (x) − g(x)] dx.