4.3. THE DEFINITE INTEGRAL
247
velocity is positive, and moving to the left when its velocity is negative. Assume that
the given velocity function is valid for t = 0 to t = 4.
1
2
3
4
-2
-1
1
2
ft/sec
sec
y = v(t)
Figure 4.29: The velocity function of a moving object.
(a) Write an expression involving definite integrals whose value is the total change
in position of the object on the interval [0, 4].
(b) Use the provided graph of v to determine the value of the total change in
position on [0, 4].
(c) Write an expression involving definite integrals whose value is the total distance
traveled by the object on [0, 4]. What is the exact value of the total distance
traveled on [0, 4]?
(d) What is the object’s exact average velocity on [0, 4]?
(e) Find an algebraic formula for the object’s position function on [0, 1.5] that
satisfies s(0) = 0.
2. Suppose that the velocity of a moving object is given by v(t) = t(t − 1)(t − 3), measured
in feet per second, and that this function is valid for 0 ≤ t ≤ 4.
(a) Write an expression involving definite integrals whose value is the total change
in position of the object on the interval [0, 4].
(b) Use appropriate technology (such as http://gvsu.edu/s/a9 9 ) to compute
Riemann sums to estimate the object’s total change in position 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.
(c) Write an expression involving definite integrals whose value is the total distance
traveled by the object on [0, 4].
9 Marc Renault, Shippensburg University.
247
velocity is positive, and moving to the left when its velocity is negative. Assume that
the given velocity function is valid for t = 0 to t = 4.
1
2
3
4
-2
-1
1
2
ft/sec
sec
y = v(t)
Figure 4.29: The velocity function of a moving object.
(a) Write an expression involving definite integrals whose value is the total change
in position of the object on the interval [0, 4].
(b) Use the provided graph of v to determine the value of the total change in
position on [0, 4].
(c) Write an expression involving definite integrals whose value is the total distance
traveled by the object on [0, 4]. What is the exact value of the total distance
traveled on [0, 4]?
(d) What is the object’s exact average velocity on [0, 4]?
(e) Find an algebraic formula for the object’s position function on [0, 1.5] that
satisfies s(0) = 0.
2. Suppose that the velocity of a moving object is given by v(t) = t(t − 1)(t − 3), measured
in feet per second, and that this function is valid for 0 ≤ t ≤ 4.
(a) Write an expression involving definite integrals whose value is the total change
in position of the object on the interval [0, 4].
(b) Use appropriate technology (such as http://gvsu.edu/s/a9 9 ) to compute
Riemann sums to estimate the object’s total change in position 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.
(c) Write an expression involving definite integrals whose value is the total distance
traveled by the object on [0, 4].
9 Marc Renault, Shippensburg University.