6. Integration
6.5: Fundamental Theorem of Calculus (continue)
Example 1. Find the area under the parabola y = −(x − 1)2 + 1 for x in the interval [0,2].
NOTE: BecauseR of the relation given by the FTC between the antiderivatives and integrals, the indefinite integral f (x) dx as a function is traditionally used for an antiderivative of f .
The connection between definite integral (as a scalar) with the indefinite integral (as a function)
is given by FTC: If f is continuous on [a, b] then
Z
b
Z b
f (x) dx =
f (x) dx .
a
a
Example 2. (Stewart) Suppose an object is moving along a straight line with position function s(t),
velocity function v(t) and acceleration function a(t). Since s0 (t) = v(t), the FTC gives
Z
t2
v(t) dx = s(t2 ) − s(t1 ),
t1
which is the displacement of the object during the time period [t1 , t2 ]. Hence, we can compute the
displacement by integration if the velocity function is known.
Similarly, since v 0 (t) = a(t), the FTC gives
Z t2
a(t) dx = v(t2 ) − v(t1 ),
t1
which is the change in velocity during the time period [t1 , t2 ].
If we want to calculate the distance traveled during the time interval, we have to consider the intervals
when v(t) ≥ 0 (the object moves to the right) and also the intervals when v(t) ≤ 0 (the object moves
to the left). In both cases, the distance is computed by integrating |v(t)|, the speed. Therefore,
Z t2
|v(t)| dx = total distance traveled.
t1
1
Example 3. An object moves along a line so that its velocity at time t is v(t) = 8t + 1 where position
is given in meters and t is given in seconds.
a) Find the displacement of the particle during the time period [0, 2].
b) Find the distance traveled during the this time period.
Finding Approximations of Definite Integrals on the Calculator: (Bollinger) If f is a continous
Rb
function on [a, b], then the value a f (x) dx can be estimated by the following command
M AT H → 9 : f nInt → f nInt(f (x), x, a, b).
Average value of a function
DEFINITION: The average value of a continuous function f on the interval [a, b] is given by
1
b−a
b
Z
f (x) dx.
a
Example 4. Find the average value of the following function on the given interval.
a) f (x) = 9 on [1,10].
b) f (x) = 2x2 + x − 5 on [-1,1].
2