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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 following materials are covered in chapter 4
...

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)
𝑑𝑥

= 𝑓(𝑥)

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
...


The following materials are covered in chapter 5
...
4 of the textbook
...


Now, let’s look at Riemann sum
...
}
The above expression means n is a natural number like 1, 2 ,3, etc
...

Let there be a closed interval [a, b]
...
…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
...

Now, let Sn = nth Riemann sum
𝑛

𝑆𝑛 = ∑ 𝑓(𝑥𝑖 )Δ𝑥
𝑖=1

Sn is a real number
...
Δ𝑥 is
the interval length
...
Addition of the
rectangles up to nth gives the nth Riemann sum
...


𝑛

𝑏

lim ∑ 𝑓(𝑥𝑖 )Δ𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 = lim 𝑆𝑛

𝑛→∞

𝑖=1

𝑎

𝑛→∞

Here, b is he right endpoint of the nth interval/segment, and a is the left endpoint of the first
interval/segment
...


A simpler explanation of the definite integral, is that it is the area under a curve, within a closed interval,
or a fixed length along the x-axis
...


To find R, or the area under the curve f(x) within the closed interval [a, b], we find the definite integral
of f(x) within the closed interval [a, b], or in other words, we find
𝑏

∫𝑎 𝑓(𝑥) 𝑑𝑥
= 𝐹(𝑏) − 𝐹(𝑎)
Here, F(x) is the antiderivative of f(x)
...


In this picture, part of the curve goes under the x-axis
...
Now the formula to find the area is as follow,
𝑏

𝑐

𝑑

∫ 𝑓(𝑥)𝑑𝑥 − ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑓(𝑥)𝑑𝑥
𝑎

𝑏

𝑐

= (F(b) - F(a)) - (F(c) - F(b)) + (F(d) - F(c))
Here, F(x) is antiderivative of f(x)
...


3

Let’s find the area under √𝑥 in the closed interval [0, 3]
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25



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