The fundamental theorem of calculus plays a key role in calculus and provides us with a connection between derivatives and integrals in two ways.
The first part states that differentiation is the opposite of integration and implies the existence of the anti-derivative for any continuous function.
The second part gives us a means to calculate the definite integral of a function over an interval by finding an anti-derivative of the function, and implies that integration is the opposite of differentiation up to the addition of a constant.
The theorem links the algebraic process of anti-differentiation to the geometrical interpretation of integration and gives us a way to calculate areas under curves for continuous functions.
Part 1:
For a function $f\left(x\right)$f(x) continuous over the interval $\left(a,b\right)$(a,b), if we define $F\left(x\right)$F(x) by the integral 
, then:
Part 2:
For a function $f\left(x\right)$f(x) continuous over a closed interval $\left[a,b\right]$[a,b] with an antiderivative function $F\left(x\right)$F(x):
We have already seen how to apply the second part of the theorem to evaluate indefinite integrals and find areas under curves. This theorem also highlights the connection between differentiation and integration as reverse processes. This property can help us simplify calculations that appear quite complex if we recognise we are taking the integral of a derivative function or if we are taking the derivative of an integral.
Worked examples
Example 1
Find $A'(x)$A′(x), given that 
.
Think: We are trying to find the derivative of an integral. If we look at the first part of the fundamental theorem of calculus we can see this will result in the function inside the integral with the variable $t$t, replaced with $x$x.
Do: $A'(x)=\frac{4x}{\sqrt{x^2-5}}$A′(x)=4x√x2−5. Let's also briefly look at how this would result from using the second part of the fundamental theorem of calculus. Let $F(t)$F(t) be an antiderivative of $f(t)$f(t), where $f(t)=\frac{4t}{\sqrt{t^2-5}}$f(t)=4t√t2−5, then we have:
| $A(x)$A(x) | $=$= | $F(x)-F(3)$F(x)−F(3) |
From the fundamental theorem of calculus |
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| $\therefore\ A'(x)$∴ A′(x) | $=$= | $F'(x)-F'(3)$F′(x)−F′(3) |
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| $=$= | $F'(x)$F′(x) |
Since the derivative of a constant is $0$0 |
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| $=$= | $f(x)$f(x) |
As $F(x)$F(x) was an antiderivative of $f(x)$f(x) |
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| $=$= | $\frac{4x}{\sqrt{x^2-5}}$4x√x2−5 |
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Reflect: As the derivative of a constant is zero, notice that the lower bound of the integral is irrelevant. That is, the exact value of the constant $a$a in the equation has no impact on the outcome.
Example 2
Determine 
.
Think: We are trying to find the derivative of an integral. The fundamental theorem of calculus tells us the result will be the function inside the integral with the variable $t$t, replaced with $x$x.
Do:
EXAMPLE 3
Calculate 
.
Think: We are trying to find the definite integral of a derivative. The process of integration will undo the process of differentiation up to the addition of a constant. So we can jump to the second step of evaluating the inner function $f(x)=\frac{x^4}{x^2+1}$f(x)=x4x2+1 at the end points of the integral and subtracting.
Do:
Practice questions
Question 1
The function $f$f has an antiderivative $F$F, and $F\left(3\right)=4$F(3)=4.
Express $\int_3^xf\left(t\right)dt$∫x3f(t)dt in terms of $F$F and $x$x.
Now, calculate $\frac{d}{dx}\int_3^xf\left(t\right)dt$ddx∫x3f(t)dt.
Question 2
Consider the expression $\frac{d}{dx}\int_k^x\frac{1}{\sqrt{t}}dt$ddx∫xk1√tdt.
What restrictions must be on the value of $k$k for the integration to be possible?
$k<0$k<0
A$k\le0$k≤0
B$k>0$k>0
C$k\ge0$k≥0
DEvaluate $\int_k^x\frac{1}{\sqrt{t}}dt$∫xk1√tdt.
Hence, evaluate $\frac{d}{dx}\int_k^x\frac{1}{\sqrt{t}}dt$ddx∫xk1√tdt and express your answer without using powers.
Question 3
Calculate the value of $\int_0^{\frac{\pi}{2}}\frac{d}{dx}\sin\left(3x-\frac{\pi}{4}\right)dx$∫π20ddxsin(3x−π4)dx.
The signed area function
In proving the fundamental theorem of calculus we considered the geometrical interpretation of the definite integral as the signed area between a curve and the $x$x-axis, and defined the signed area function as:
Specifically, given a function $f\left(t\right)$f(t) which is continuous on the interval $a\le t\le b$a≤t≤b. We define the signed area function $A\left(x\right)$A(x) to be the value of the net area under the curve $f\left(t\right)$f(t) between $t=a$t=a and $t=x$t=x, where $a\le x\le b$a≤x≤b.
For a positive function this will represent the shaded area shown below:
For a function which lies completely or partially below the horizontal axis, the value of the signed area function represents the total area above the horizontal axis minus the total area below the horizontal axis. For example, for the function below $A(x)=A_1-A_2$A(x)=A1−A2.
From the second part of the fundamental theorem of calculus we have that the signed area function for a function $f\left(t\right)$f(t) continuous over a closed interval $\left[a,b\right]$[a,b] with an antiderivative function $F\left(t\right)$F(t), can be evaluated as:
Worked example
Example 4
Consider the function $f(t)=8-2t$f(t)=8−2t for $t\ge0$t≥0 and the signed area function 
.
(a) State the function $A(x)$A(x) in terms of $x$x.
Think: Use the fundamental theorem of algebra to evaluate the the integral.
Do:
(b) Hence, evaluate $A(3)$A(3) and give a geometrical interpretation.
Think: Substitute $x=3$x=3 into the formula for $A(x)$A(x).
Do:
| $A(3)$A(3) | $=$= | $8(3)-(3)^2$8(3)−(3)2 |
| $=$= | $15$15 |
Interpretation: This will be the value of the signed area from $t=0$t=0 to $t=3$t=3, as the graph is above the horizontal axis for this interval this can be interpreted as the area shown below and could be confirmed by finding the area geometrically. Thus, the shaded area is $15$15 units2.
(c) For what values of $x$x is the function $A(x)$A(x) increasing?
Think: Visualise the area formed under the graph as $x$x increases from zero. This area will continue to grow until the graph falls below the horizontal axis and the area under the curve is subtracted from the area above.
Do: The function will increase until $x=4$x=4. Thus, the function is increasing on the interval $0
(d) Find when $A(x)=0$A(x)=0 and give a geometrical interpretation of this.
Think: Solve for when $A(x)=0$A(x)=0 and visualise this value on the graph.
Do: If $A(x)=0$A(x)=0, we have:
| $8x-x^2$8x−x2 | $=$= | $0$0 |
| $x(8-x)$x(8−x) | $=$= | $0$0 |
| $x$x | $=$= | $0$0 or $8$8 |
Interpretation: For $x=0$x=0 we would have 
which indicates the area under the curve from $t=0$t=0 to itself. Thus, the result indicates the area under a point is zero.
For $x=8$x=8, we can see from the graph below that the function gives the result of the signed area from $t=0$t=0 to $t=8$t=8, from the resulting net area of zero we can conclude that $A_1=A_2$A1=A2.
Practice questions
Question 4
Consider the function $f\left(t\right)=2t$f(t)=2t, where $t\ge0$t≥0.
Let $A\left(x\right)$A(x) be the area function which represents the area bound by $f\left(t\right)$f(t) and the horizontal axis from $0$0 to $x$x. That is, $A\left(x\right)=\int_0^xf\left(t\right)dt$A(x)=∫x0f(t)dt.
Find an expression for $A\left(x\right)$A(x).
Hence determine the area under the function $f\left(t\right)$f(t) from $t=0$t=0 to $t=4$t=4.
Question 5
Given that $F\left(x\right)=\int_1^xf\left(t\right)dt$F(x)=∫x1f(t)dt, with $F\left(4\right)=78$F(4)=78 and $F''\left(x\right)=6x$F′′(x)=6x, find $f\left(t\right)$f(t).
Question 6
Consider the function $f\left(t\right)=2t-6$f(t)=2t−6, and the signed area function $A\left(x\right)=\int_0^xf\left(t\right)dt$A(x)=∫x0f(t)dt.
Complete the following table of values:
$x$x $0$0 $1$1 $2$2 $3$3 $4$4 $5$5 $6$6 $7$7 $A\left(x\right)$A(x) $0$0 $-5$−5 $\editable{}$ $\editable{}$ $\editable{}$ $-5$−5 $\editable{}$ $7$7 For what values of $x$x is the function $A\left(x\right)$A(x) increasing?
Give your answer as an inequality.
Sketch a graph of $y=A\left(x\right)$y=A(x) for $0\le x\le7$0≤x≤7 below.
Loading Graph...At what values of $x$x is $A\left(x\right)=0$A(x)=0?
Enter both values on the same line, separated by a comma.
What is the significance of these values of $x$x?
These values are the points where the function $f\left(t\right)$f(t) crosses the $x$x-axis.
AThe gradient of the curve $f\left(t\right)$f(t) is horizontal at both of these points.
BThere is no area under the curve $f\left(t\right)$f(t) at either of these points.
CAt these $x$x-values, there are equal amounts of area on either side of the $x$x-axis under the curve $f\left(t\right)$f(t).
DState the function $A\left(x\right)$A(x) in terms of $x$x.