THE FUNDAMental THEOREM OF CALCULUS ILLUSTRATIONS OF THE FUNDAMENTAL THEOREM In this section, we will illustrate both parts of the Fundamental Theorem by applying them in solving various integrals. Examples 1 through 12 will deal with Part 1 of the Theorem, while examples 13 to 19 will deal with Part 2. To be able to evaluate an integral using FTC 2, you need to know how to compute an antiderivative for any given function. The table below lists some general functions and their corresponding antiderivative formulas: Function General Antiderivative f(x) + g(x) F(x) + G(x) cf(x) cF(x) cos x sin x sec 2 x tan x sec x tan x sec x x n (n ≠ -1) Study the following examples carefully. Example 1 Using Part 1 of the Fundamental Theorem, compute the derivative of Solution Always remember this: to differentiate an integral, the first step should be to ensure that the integrand is a continuous function; because a discontinuous function IS NOT differentiable. Thus, since the integrand f(t) = 1 + 2t is continuous, the FTC 1 gives: Example 2 Using Part 1 of the Fundamental Theorem, compute the derivative of Solution The integrand f(t) = (2 + t 4 ) 5 is continuous. Therefore, using FTC 1 gives its derivative as: x n+1 n + 1 g(x) = √ 1 + 2t dt ∫ 0 x g '(x) = √ 1 + 2x g(x) = (2 + t 4 ) 5 dt ∫ 1 x g'(x) = (2 + x 4 ) 5
√1 + 2t dt Function General Antiderivative √1 + 2x ∫ 1 ∫ 0 g(x) = g(x) = g '(x) = x n+1 n + 1 g'(x) = (2 + x 4 ) 5 In this section, we will illustrate both parts of the Fundamental Theorem by applying them in solving various Always remember this: to differentiate an integral, the first step should be to ensure that the integrand is a To be able to evaluate an integral using FTC 2, you need to know how to compute an antiderivative for any given Study the following examples carefully. x
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THE FUNDAMental THEOREM OF CALCULUSILLUSTRATIONS OF THE FUNDAMENTAL THEOREM
In this section, we will illustrate both parts of the Fundamental Theorem by applying them in solving various
integrals. Examples 1 through 12 will deal with Part 1 of the Theorem, while examples 13 to 19 will deal with Part 2.
To be able to evaluate an integral using FTC 2, you need to know how to compute an antiderivative for any given
function. The table below lists some general functions and their corresponding antiderivative formulas:
Function General Antiderivative
f(x) + g(x) F(x) + G(x)
cf(x) cF(x)
cos x sin x
sec 2 x tan x
sec x tan x sec x
xn (n ≠ -1)
Study the following examples carefully.
Example 1Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionAlways remember this: to differentiate an integral, the first step should be to ensure that the integrand is a
continuous function; because a discontinuous function IS NOT differentiable. Thus, since the integrand f(t) = 1 + 2t
is continuous, the FTC 1 gives:
Example 2Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionThe integrand f(t) = (2 + t4)5 is continuous. Therefore, using FTC 1 gives its derivative as:
xn+1
n + 1
g(x) = √1 + 2t dt∫ 0
x
g '(x) = √1 + 2x
g(x) = (2 + t4)5 dt∫ 1
x
g'(x) = (2 + x4)5
Example 3Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionThe integrand f(t) = t2 sin t is continuous; therefore,
Example 4Using Part 1 of the Fundamental Theorem, compute the derivative of
Solution The integrand f(x) = [1/(x + x2)] is a continuous function on the given interval. Therefore, FTC 1 gives the
derivative of g as
Example 5Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionHere, the integrand is F(t) = cos(t2). This function is continuous, we can therefore obtain a derivative. However,
there is one small problem: recall that the task is to differentiate the integral function F with respect to x which is
supposed to be the upper limit of the integral. But in this case, x has become the lower limit. Thus, to move any
further, what we need to do is “tweak” the integral so that x becomes the upper limit. To do this, we apply property 1
of definite integrals, so that
Thus, we'll be dealing with
Based on FTC 1, the derivative of F is
g(y) = (t2 sin t) dt∫ 2
y
g'(y) = y2 sin y
g(u) = [1/(x + x2)] dx∫ 3
u
g'(u) = 1/(u + u2)
F(x) = [cos(t2)] dt∫ x
2
F(x) = [cos(t2)] dt∫ x
2 = [cos(t2)] dt∫ 2
x–
F(x) = [cos(t2)] dt∫ 2
x–
F'(x) = – cos(x2)
Example 6Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionThis is pretty much like example 5 above; we have to apply property 1 of the definite integral before computing the
derivative of the integral. Using property 1 of integrals, F becomes
Thus, since F is continuous, its derivative is
Example 7Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionFrom the integral above, we find that the integrand is
F(t) = sin4 t
Observe the upper limit of the integral: 1/x. The upper limit is a function on its own. Thus, what we have here is a
function within a function, a situation which usually calls for the chain rule which, is widely used in differentiating
composite functions.
The first step to differentiating functions like this is to treat the upper limit as a function, and we do that by
representing it appropriately as such. Thus, let's assume
m = 1/x [or m(x) = 1/x)]This gives
So now, our task is to find
This is the point where the chain rule is put into use; we'll be differentiating two functions: the integral itself and the
function representing the upper limit. For the integral, its derivative is
F(x) = [tan θ] dθ∫ x
10
F(x) = [tan θ] dθ∫ 10
x–
F'(x) = – tan x
h(x) = [sin4t] dt∫ 2
1/x
h(x) = [sin4t] dt∫ 2
m
h'(x) = [sin4t] dt∫ 2
mEQUATION 1
d
dm
h'(x) = [sin4t] dt∫ 2
md
dx
and the derivative of the upper limit is
Using the chain rule, we put both equations 1 and 2 together so that
h'(x) = sin4 m × (-x-2)
Example 8Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionThe integrand f(r) = √1+r3 is continuous. Again observe the upper limit; it is a function on its own. We will therefore
have to make use of the chain rule here. Let's assume
b = x2 or b(x) = x2
Therefore,
h'(x) = √1 + b3 × 2x
h'(x) = 2x √1 + (x2)3
h'(x) = 2x √1 + x6
EQUATION 2m' (x)dm
dx= =
d
dx(1/x)
h'(x) = [sin4t] dt∫ 2
m dm
dx×
d
dm
h' (x) = 1x
sin4 × (-x-2)
1x2 –
1x
sin4h' (x) =
x2
– 1x
sin4
h' (x) =
h(x) = √1 + r3 dr∫ 2
x2
h'(x) = [√1 + r3] dr∫ 0
b d
dx
h'(x) = [√1 + r3] dr∫ 0
b d
db
db
dx×
Example 9Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionThe integrand f(t) = (cos t)/t is continuous on the given interval. The upper limit in this case is the root function
√ x. Therefore, let
m = √x or m(x) = √x
Thus,
Example 10Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionThe integrand here is f(t) = t + sin t which is continuous on the given interval. Let w represent the upper limit, i.e.
w = cos x or w(x) = cos x
Therefore,
y = dt∫ 3
√x cos tt
y' = ∫ 3
m d
dm
y ' = dt∫ 3
√x cos tt
ddx
dtcos tt
dm
dx×
×cos mm
1
2√xy' =
×cos √x√x
1
2√xy' =
cos √x
2xy' =
y = [t + sin t] dt∫ 1
cos x
y ' = [t + sin t] dt∫ 1
cos xddx
y ' = [t + sin t] dt∫ 1
wddw
dw
dx×
y' = (w + sin w) × - sin x
y' = - sin x (w + sin w)
y' = - sin x(cos x + sin [cos x])
y' = - sin x cos x – sin x sin(cos x)
Example 11Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionBy property 1 of definite integrals, we have
Let a(x) = 1 – 3x. Thus, we have
Hence, the derivative of y equals
Since a = 1 – 3x, then,
Example 12Using Part 1 of the Fundamental Theorem, compute the derivative of
SolutionUsing property 1 of definite integrals, we have
y = du∫ 1 -3x
1 u3
1 + u2
y = du∫ 1
1 -3x u3
1 + u2–
y = du∫ 1
a u3
1 + u2–
y ' = du∫ 1
a u3
1 + u2– d
dada
dx×
- 3×– a3
1 + a2y ' =
3a3
1 + a2y ' =
3(1 – 3x)3
1 + (1 – 3x)2y ' =
y = [sin 3t] dt∫ 1/x2
0
Let
m(x) = 1/x2 = x-2
Thus, we have
Hence, the derivative y' equals
y' = - sin3 m × -2x-3
In the next set of examples, we will illustrate Part 2 of the Fundamental theorem by using it to evaluate different
kinds of definite integrals
Example 13Evaluate the integral using Part 2 of the Fundamental Theorem, or explain why it doesn't exist.
SolutionTo evaluate an integrals using FTC 2, the first step to be taken, like in using FTC 1, is to ensure that the integrand is
continuous on that particular interval. In this case, the integrand is the function y = f(x) = x5 and it's continuous on
the interval [1, 3]. Next, we have to figure out an antiderivative for the integrand.
Using the general antiderivative formula
(Where c is an arbitrary constant)
We find that the derivative of f in this case is
F(x) = x6/6
Recall the FTC 2 formula:
y = [sin 3t] dt∫ 0 1/x2
–
y = [sin 3t] dt∫ 0m
–
y ' = [sin 3t] dt∫ 0m
– ddm
dm
dx×
y' = 1x2–sin3 × (-2x-3)
x3
y' =
1x2
2 sin3
x5 dx∫ -1
3
x n+1
n + 1F(x) = + C
f(x) dx∫ a
b= F(b) – F(a)
This means the antiderivative will be evaluated at the two values a and b. In this case, a = -1 and b = 3. Therefore,
= [(3)6/6] - [(-1)6/6]
= (729/6) - (1/6)
= 728/6
= 364/3 Hence
Example 14Evaluate the integral using Part 2 of the Fundamental Theorem, or explain why it doesn't exist.
SolutionThe integrand f(x) = 4x + 3 is continuous on the interval [2, 8]. Using the general antiderivative formula, we find
that the antiderivative F of f is
F(x) = 2x2 + 3x
Therefore, Part 2 of the fundamental theorem gives
(where a = 2 and b = 8)
Example 15
Evaluate the integral using Part 2 of the Fundamental Theorem, or explain why it doesn't exist.
SolutionHere, we have the integrand f(y) = 1 + 3y - y2 , whose antiderivative is given by
f(x) dx∫ a
b= F(b) – F(a)
x5 dx∫ -1
3 = 364/3 ≈ 121.3
(4x + 3) dx∫ 2
8
(4x + 3) dx∫ 2
8= F(b) – F(a)
(4x + 3) dx∫ 2
8 = [2(8)2 + 3(8)] - [2(2)2 + 3(2)]
= (128 + 24) - (8 + 6)
= 152 - 14
= 138
(1 + 3y – y2) dy∫ 0
4
F(y) = 3y2
2y + –
y3
3
Therefore, Part 2 of the fundamental theorem gives
(where a = 0 and b = 4)
= 4 + 24 – (64/3) - 0
= 20/3Therefore,
Example 16Evaluate the integral using Part 2 of the Fundamental Theorem, or explain why it doesn't exist.
SolutionFor ease of simplification, this integral can be better expressed as
Using the general antiderivative formula, the antiderivative of the integrand becomes
F(t) = -t-3 = -1/t3
Using FTC 2, we have
(Where a = -1 and b = 1)
Hence, the value of the integral is -2. Or is it!!!!
So far, it seems we've performed the calculation correctly and obtained the right answer.
(1 + 3y – y2) dy∫ 0
4= F(b) – F(a)
(1 + 3y – y2) dy∫ 0
4 = –3(4)2
2(4) + –
(4)3
33(0)2
2(0) + –
(0)3
3
(1 + 3y – y2) dy∫ 0
4= 20/3 ≈ 6.667
(3/t4) dt∫ -1
1
= F(b) – F(a) (3t–4) dt∫ -1
1
(3t–4) dt∫ -1
1= [-1/(1)3] – [-1/(-1)3]= (-1/1) - (-1/-1)
= -1 - 1
= - 2
(3t–4) dt∫ -1
1
Illustration 1: Graph of y = 3/t4. Observe the discontinuity in the interval [-1, 1]
There is however, one small problem: THIS ANSWER IS WRONG. The reason for this is because the integrand f(t)
= 3t-4 is NOT CONTINOUS ON THE GIVEN INTERVAL [-1, 1]. The graph above clearly shows the discontinuity.
This is why it is specifically mentioned in the Fundamental Theorem; to be able to evaluate or differentiate an
integral, the function (integrand) concerned MUST BE CONTINUOUS ON THE INTERVAL [a, b]. Therefore the
correct answer should be
Example 17Evaluate the integral using Part 2 of the Fundamental Theorem, or explain why it doesn't exist.
SolutionThe integrand f(θ) = cosθ is continuous on the interval [π, 2π]. The antiderivative formula gives F as
F(θ) = sin θ Using FTC 2, we have
DOES NOT EXIST (3t–4) dt∫ -1
1
(cos θ) dθ∫ π
2π
(cos θ) dθ∫ π
2π= F(b) – F(a)
(where a = π and b = 2π). So,
If you use a calculator or computer you might get something like 5.91991876344e-13 which is indeed very close to
zero.
Example 18Evaluate the integral using Part 2 of the Fundamental Theorem, or explain why it doesn't exist.
SolutionWe are about to evaluate the integral of a piecewise defined function. One good way to start is by graphing the
function f so as to get a visual of the area we are looking for. That way, we'll know how to approach the problem. The
graph is drawn above. Notice how both functions seem to form one continuous line. This is because both functions
have the same value when x = 1. Thus, both functions join at that point (indicated by the black dot).
Illustration 2: Graphs of y = x4 (red line) and y = x5 (blue line). Notice that both functions have the same value when
x = 1. They therefore seem to form one continuous line.
The graph below shows the areas bounded by each graph. The yellow area represents the area under the function y
= x5 on the interval [1, 2], while the small, grey area represents the area under the function y = x4 on the interval [0,
1]:
(cos θ) dθ∫ π
2π= sin 2π - sin π
= 0
x4 if 0 ≤ x < 1 = f(x) dx∫ 0
2
x5 if 1 ≤ x ≤ 2
From the graph, it is evident that we will have to find two areas:
which happen to be adjacent to each other. Thus, the sum of the areas will represent the value of the integral
This is simply another way of illustrating property 7 of the definite integral. Using FTC 2,
(where a = 0, b = 1 and the antiderivative F(x) = 0.2x5). Thus,
AND
(where a = 1, b = 2 and the antiderivative F(x) = x6/6). Thus,
Hence,
AND (x4) dx∫ 0
1 (x5) dx∫ 1
2
f(x) dx∫ 0
2
(x4) dx∫ 0
1= F(b) – F(a)
(x4) dx∫ 0
1= F(1) – F(0) = 0.2(1)5 - 0.2(0)5
= 0.2
(x5) dx∫ 1
2= F(b) – F(a)
(x5) dx∫ 1
2 = [(2)6/6] - [(1)6/6]
= 64/6 - 1/6 = 63/6 = 10.5
f(x) dx∫ 0
2= (x4) dx∫ 0
1 (x5) dx∫ 1
2+
= 0.2 + 10.5 = 10.7
Example 19Evaluate the integral using Part 2 of the Fundamental Theorem, or explain why it doesn't exist.
SolutionLike example 18 above, we start by graphing f:
A graph of the piecewise function y = f(x). The orange area represents the area under the function y = sin x on the
interval [0, π], while the green area represents the area under the function y = x on the interval [-π, 0]
From the graph, we'll be dealing with two areas:
Also, from the graph, we find that
Let's deal with each integral one at a time. Thus, we have
(where a = 0, b = π and the antiderivative F(x) = -cos x). Therefore,
x if -π ≤ x ≤ 0= f(x) dx∫ -π
π
sin x if 0 ≤ x ≤ π
AND (x) dx∫ -π
0 (sin x) dx∫ 0
π
+ f(x) dx∫ -π
π (sin x) dx∫ 0
π (x) dx∫ -π
0= –
f(x) dx∫ -π
π (sin x) dx∫ 0
π (x) dx∫ -π
0= –
(sin x) dx∫ 0
π= F(b) – F(a)
AND
(where a = -π, b = 0 and F(x) = x2/2). Therefore,
Hence,
So,
Unlike example 18, this area is a result of the difference between adjacent areas.
Example 20Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact
area.
y = 3√x , 0 ≤ x ≤ 27
SolutionUsing the definite integral notation, we can express the required area as
But first, let's estimate the area using a graph (which will involve the Riemann method. This graph shows the
required area represented by the integral above, divided into 9 equal sub areas.
From the graph, we find that a = 0, b = 27 and n = 9. Therefore,