Definite Integrals

Name: Definite Integrals
Author: StemPy Lab

How much distance does a bus cover if you know its speed at every moment? The answer is the area under the speed graph. Meet the area function, see why its derivative is the original function, and use the Fundamental Theorem of Calculus to evaluate definite integrals. Then use substitution and the properties of definite integrals, and see the integral as the limit of a sum. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
The area function and the Fundamental Theorem of Calculus
The definite integral $\int_a^b f(x)\,dx$ is the signed area between the graph of $f$ and the $x$ -axis from $a$ to $b$ . The area function $A(x) = \int_a^x f(t)\,dt$ has the derivative $f(x)$ . This is the first part of the Fundamental Theorem of Calculus. The second part says $\int_a^b f(x)\,dx = F(b) - F(a)$ for any antiderivative $F$ of $f$ .
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
Substitution and the properties of definite integrals
In a definite integral, a substitution $u = g(x)$ also changes the limits, so there is no need to return to $x$ . The properties of definite integrals let us split an interval, use the symmetry of even and odd functions, and replace $x$ by $a + b - x$ . These properties give exact answers to integrals that have no antiderivative in simple form.
1 simulation1 practice question1 Python program4 problems
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Lesson 3about 25 min
The definite integral as the limit of a sum
Divide $[a, b]$ into $n$ equal strips of width $h = \dfrac{b - a}{n}$ and add the areas of the rectangles. The sum $h\sum f(a + rh)$ tends to $\int_a^b f(x)\,dx$ as $n \to \infty$ . With the formulas for $\sum r$ , $\sum r^2$ and $\sum r^3$ we can find an integral from this definition. Read in the other direction, a limit of a sum such as $\lim \dfrac{1}{n}\sum f\left(\dfrac{r}{n}\right)$ is the integral $\int_0^1 f(x)\,dx$ .
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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Definite Integrals: Chapter Test

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Definite Integrals

Course outline (3 lessons)

The area function and the Fundamental Theorem of Calculus

Substitution and the properties of definite integrals

The definite integral as the limit of a sum

Test Your Learning

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