The fundamental theorem of calculus
A Luas tram's speed is recorded every second, and you want the distance it travels between two stops. Adding up the distance in tiny steps is slow. The Fundamental Theorem of Calculus links the integral (area) with the derivative (slope) and gives you a quick method: evaluate G(b) − G(a). By the end of this page you will use it to turn a speed into a distance.
The function defined by an integral
Fix the lower limit $a$ and let the upper limit vary: for each $x$, $F(x) = \int_a^x f(t)\,dt$ is the area accumulated from $a$ to $x$. $F$ is a new function, with one value for each $x$.
Two properties of definite integrals will be useful: $\int_a^b k f(x)\,dx = k\int_a^b f(x)\,dx$ and $\int_a^c f(x)\,dx + \int_c^b f(x)\,dx = \int_a^b f(x)\,dx$.
Given a function $f(x)$ continuous on the interval $[1, 8]$, the values of the following definite integrals are known: $\int_1^5 f(x) \, dx = 7$…
Given a function $f(x)$ continuous on the interval $[1, 8]$, the values of the following definite integrals are known:
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$\int_1^5 f(x) \, dx = 7$
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$\int_5^8 f(x) \, dx = -3$
Use the properties of the definite integral to calculate the value of $\int_1^8 2 \times f(x) \, dx$.
The fundamental theorem of calculus
This is the key result that links the whole unit: the derivative of this accumulated area function is the original function.
When the upper limit is not $x$ but a function of $x$ (for example $x^2$), you must also apply the chain rule.
When the lower limit also varies, split the integral at a constant $c$: $\int_{h(x)}^{g(x)} f = \int_c^{g(x)} f - \int_c^{h(x)} f$, so the derivative is $f(g)g' - f(h)h'$.
When the integrand has no elementary antiderivative, you still know $F'(x)$. The sign of $f$ tells you where $F$ increases or decreases, and $F''(x)=f'(x)$ gives the concavity.
A video game programmer is designing the trajectory of a projectile. The velocity of the object as a function of time $t$ is given by the expression…
A video game programmer is designing the trajectory of a projectile. The velocity of the object as a function of time $t$ is given by the expression $v(t) = \int_1^t (3x^2 - 4x) \, dx$.
Apply the Fundamental Theorem of Calculus to find the expression for the instantaneous acceleration $a(t)$, knowing that acceleration is the derivative of velocity with respect to time ($a(t) = v'(t)$).
Find the value of the derivative of the following integral function at the point $x = 2$: $$F(x) = \int_{0}^{x} \sqrt{t^3 + 1} \, dt$$ Explain which mathematical theorem…
Find the value of the derivative of the following integral function at the point $x = 2$:
$$F(x) = \int_{0}^{x} \sqrt{t^3 + 1} \, dt$$Explain which mathematical theorem you are using to solve the problem without needing to calculate the antiderivative.
Calculate the derivative function $F'(x)$ of the following integral function using the Fundamental Theorem of Calculus and the chain rule: $$F(x) = \int_{1}^{x^2}…
Calculate the derivative function $F'(x)$ of the following integral function using the Fundamental Theorem of Calculus and the chain rule:
$$F(x) = \int_{1}^{x^2} \frac{\ln(t)}{t + 1} \, dt$$State the domain of definition of the function $F(x)$ so that the integral is well-defined.
Given the integral function $H(x) = \int_{0}^{x} \frac{t-1}{t^2+4} \, dt$: a) Find the intervals of increase and decrease of $H(x)$. b) Determine the coordinates of the…
Given the integral function $H(x) = \int_{0}^{x} \frac{t-1}{t^2+4} \, dt$:
- a) Find the intervals of increase and decrease of $H(x)$.
- b) Determine the coordinates of the local extrema (maximums or minimums) of $H(x)$ and classify them.
Let the function $f(t) = \frac{\ln(t)}{t^2}$ be defined for $t > 0$. A new integral function $G(x)$ is defined as: $$G(x) = \int_{x}^{x^2} f(t) \, dt$$ a) Applying the…
Let the function $f(t) = \frac{\ln(t)}{t^2}$ be defined for $t > 0$. A new integral function $G(x)$ is defined as:
$$G(x) = \int_{x}^{x^2} f(t) \, dt$$- a) Applying the Fundamental Theorem of Calculus and the chain rule, obtain the expression for the derivative $G'(x)$.
- b) Determine the intervals of increase and decrease of $G(x)$ in its domain.
- c) Find the value of $x > 1$ for which the function $G(x)$ reaches a local maximum.
Evaluating definite integrals
The fundamental theorem says that a definite integral can be calculated with any antiderivative: you evaluate the antiderivative at the two endpoints and subtract. You no longer need sums of rectangles.
When a limit has the form $\frac{0}{0}$, differentiate the numerator and the denominator separately and take the limit again (L'Hôpital's rule). This is not on the Leaving Certificate syllabus, so it is only a tool here.
To evaluate $\int_a^b f(g(x))\,g'(x)\,dx$, substitute $u=g(x)$: then $du=g'(x)\,dx$ and the limits become $g(a)$ and $g(b)$.
Some integrands, such as $\frac{1}{1+t^4}$, have no simple antiderivative, so evaluating F(b) − F(a) is of no use. Then you can still work with the fundamental theorem and L'Hôpital's rule, as above, without ever finding the antiderivative.
For the area, integrate $|f|$: split at the roots and take the absolute value of each part. An odd function ($f(-x)=-f(x)$) integrates to $0$ over $[-a,a]$. The average value of $f$ on $[a,b]$ is $\frac{1}{b-a}\int_a^b f(x)\,dx$.
A solar thermal panel has a design whose upper cross-section is bounded by the function $f(x) = 3x^2 + 2$, where $x$ is measured in metres. Calculate the value of the…
A solar thermal panel has a design whose upper cross-section is bounded by the function $f(x) = 3x^2 + 2$, where $x$ is measured in metres. Calculate the value of the definite integral on the interval $[0, 2]$ using Barrow's Rule.
Without doing complex integration calculations, justify with reasoning why the value of the following integral is equal to $0$, based on the properties of odd functions…
Without doing complex integration calculations, justify with reasoning why the value of the following integral is equal to $0$, based on the properties of odd functions and the symmetry of the interval:
$$\int_{-2}^{2} (x^5 + 3x^3 - x) \, dx$$The figure shows the graph of $f(x)=x^5+3x^3-x$ on the interval $[-2,\,2]$, with the regions between the curve and the $X$-axis shaded.
Given the function $f(x) = |x^2 - 2x|$, calculate the value of the definite integral on the interval $[0, 3]$ using Barrow's Rule: $$\int_0^3 |x^2 - 2x| \, dx$$ Briefly…
Given the function $f(x) = |x^2 - 2x|$, calculate the value of the definite integral on the interval $[0, 3]$ using Barrow's Rule:
$$\int_0^3 |x^2 - 2x| \, dx$$Briefly explain why it is necessary to split the interval of integration to solve this exercise.
Find the mean value of the function $f(x) = \sqrt{x+1}$ on the interval $[0, 3]$ and determine the point $c \in (0, 3)$ whose existence is guaranteed by the Mean Value…
Find the mean value of the function $f(x) = \sqrt{x+1}$ on the interval $[0, 3]$ and determine the point $c \in (0, 3)$ whose existence is guaranteed by the Mean Value Theorem for Definite Integrals. Interpret the result geometrically.
Given the function $f(x) = x^2 - 4$, consider the integral function $H(x) = \int_{0}^{x} |f(t)| \, dt$. a) Express $H(x)$ as a piecewise function for values of…
Given the function $f(x) = x^2 - 4$, consider the integral function $H(x) = \int_{0}^{x} |f(t)| \, dt$.
- a) Express $H(x)$ as a piecewise function for values of $x \in [0, 5]$.
- b) Study the differentiability of $H(x)$ at the point $x = 2$. Is the Fundamental Theorem of Calculus directly applicable at that point?
- c) Use the Fundamental Theorem of Calculus (evaluating the definite integral) to calculate the exact value of $H(3)$.