University of Toronto Scarborough
MATA36
Calculus II for Physical Sciences
Text Book: Calculus Early Transcendentals 9th Edition by James
Stewart, Daniel Clegg, Saleem Watson
Introduction
The course starts with a review of materials related to antiderivatives or integration from the
prerequisite course MATA30(Calculus I for Physical Sciences). The required review materials are also
covered in various chapters of the mentioned textbook.
The following materials are covered in chapter 4.9 of the textbook.
Let f be a function
Let F be an antiderivative of function f
Which means, the derivative of F is f,
⇒ F′(x) = f(x)
𝑑
⇒ 𝐹(𝑥) = 𝑓(𝑥)
𝑑𝑥
The notation for the antiderivative of f(x) is, ∫ 𝑓(𝑥)𝑑𝑥
For example,
1
∫ 𝑓(𝑥)𝑑𝑥 = ∫ √𝑥 𝑑𝑥 = ∫ 𝑥 2 𝑑𝑥
3
2𝑥 2
= 3
+𝐶
= 𝐹(𝑥)
Now, the derivative of F(x) is f(x),
𝑑 2 3
( 𝑥 2 + 𝐶)
𝑑𝑥 3
2 3 (3−1)
= × 𝑥 2
3 2
1
= 𝑥2
, = √𝑥
Another example,
∫ cos(𝑡) 𝑑𝑡 = sin(𝑡) + 𝐶
𝑑
(sin(𝑡) + 𝐶) = cos (𝑡)
𝑑𝑥
Functions like polynomial, rational, exponential, logarithmic, radical, trigonometric and the inverses of
these functions will be dealt with in the course. Not all functions have antiderivatives, but most do.
The following materials are covered in chapter 5.1 - 5.4 of the textbook.
Let f(x) be a continuous function in a closed interval [a, b],
Continuous means,
lim 𝑓(𝑥) = 𝑓(𝐶)
𝑥→𝐶
∀𝐶 ∈ [𝑎, 𝑏]
The above means that between the x-value of ‘a’ and x-value of ‘b’, as x approaches C, the limit is equal
to the y-value of the function at the x-value of C, which means that the function f(x) is continuous
without any kind of discontinuity in the closed interval [a, b].
Now, let’s look at Riemann sum. Let 𝑛 ∈ ℕ = {1, 2, 3,….}
The above expression means n is a natural number like 1, 2 ,3, etc..
The symbol ‘ℕ’ means natural number.
Let there be a closed interval [a, b].
𝑏−𝑎
Let Δ𝑥 =
𝑛
Let i = 1,2,..…n
Let the right-hand endpoint of the ith subinterval be xi
Then, xi = a + i(Δ𝑥)
The above information, on a curve f(x), is shown in picture below.
MATA36
Calculus II for Physical Sciences
Text Book: Calculus Early Transcendentals 9th Edition by James
Stewart, Daniel Clegg, Saleem Watson
Introduction
The course starts with a review of materials related to antiderivatives or integration from the
prerequisite course MATA30(Calculus I for Physical Sciences). The required review materials are also
covered in various chapters of the mentioned textbook.
The following materials are covered in chapter 4.9 of the textbook.
Let f be a function
Let F be an antiderivative of function f
Which means, the derivative of F is f,
⇒ F′(x) = f(x)
𝑑
⇒ 𝐹(𝑥) = 𝑓(𝑥)
𝑑𝑥
The notation for the antiderivative of f(x) is, ∫ 𝑓(𝑥)𝑑𝑥
For example,
1
∫ 𝑓(𝑥)𝑑𝑥 = ∫ √𝑥 𝑑𝑥 = ∫ 𝑥 2 𝑑𝑥
3
2𝑥 2
= 3
+𝐶
= 𝐹(𝑥)
Now, the derivative of F(x) is f(x),
𝑑 2 3
( 𝑥 2 + 𝐶)
𝑑𝑥 3
2 3 (3−1)
= × 𝑥 2
3 2
1
= 𝑥2
, = √𝑥
Another example,
∫ cos(𝑡) 𝑑𝑡 = sin(𝑡) + 𝐶
𝑑
(sin(𝑡) + 𝐶) = cos (𝑡)
𝑑𝑥
Functions like polynomial, rational, exponential, logarithmic, radical, trigonometric and the inverses of
these functions will be dealt with in the course. Not all functions have antiderivatives, but most do.
The following materials are covered in chapter 5.1 - 5.4 of the textbook.
Let f(x) be a continuous function in a closed interval [a, b],
Continuous means,
lim 𝑓(𝑥) = 𝑓(𝐶)
𝑥→𝐶
∀𝐶 ∈ [𝑎, 𝑏]
The above means that between the x-value of ‘a’ and x-value of ‘b’, as x approaches C, the limit is equal
to the y-value of the function at the x-value of C, which means that the function f(x) is continuous
without any kind of discontinuity in the closed interval [a, b].
Now, let’s look at Riemann sum. Let 𝑛 ∈ ℕ = {1, 2, 3,….}
The above expression means n is a natural number like 1, 2 ,3, etc..
The symbol ‘ℕ’ means natural number.
Let there be a closed interval [a, b].
𝑏−𝑎
Let Δ𝑥 =
𝑛
Let i = 1,2,..…n
Let the right-hand endpoint of the ith subinterval be xi
Then, xi = a + i(Δ𝑥)
The above information, on a curve f(x), is shown in picture below.
