Ch5 Integrals

Question 1

To find the indefinite integral is to

Answer 1

‘find the antiderivative of f(x).’ The result is a function which always includes +C

‘Find a function whose derivative with respect to x is f(x).’

Question 2

Definite Integral

Answer 2

The net area under a curve f(x) from x=a to x=b. Result is a number

Question 3

Reimann Sum

Answer 3

As n approaches infinity (the number of rectangles under the curve) and the width of each interval gets thinner, we approach the integral.

Question 4

Reverse Power Rule

Answer 4

Use when integrating an expression of the form x^n. Use if n is not -1.

Question 5

Indefinite Integrals Sums & Multiples Rule

Answer 5

You can split the integral across the sum or difference of terms.

Question 6

Constant Multiple Rule

Answer 6

You can pull constants outside the integral.

Question 7

What is the integral of 1/x?

Answer 7

Used when you have x^-1. You cannot use the reverse power rule here since n=-1, so you simplify it to be 1/x.

Question 8

Integral of e^x

Answer 8

Question 9

Integral of a^x

Answer 9

Question 10

The Fundamental Theorem of Calculus Part 1. State what it is and what is its significance.

Answer 10

If 𝑦 = 𝑓(𝑥) is continuous over an interval (𝑎, 𝑏), then the function 𝐹(𝑥) = ∫ 𝑓(𝑡) 𝑑𝑡, which represents the accumulated area under the curve from
𝑎 to 𝑥, is differentiable on (𝑎,𝑏), and its derivative is
𝐹′(𝑥)=𝑓(x). That is, the rate of change of the accumulated area equals the value (height) of the original function at 𝑥

‘The instantaneous rate of change of the area (i.e., the derivative of 𝐹(𝑥) is simply how tall the function is at that point — because that’s how fast new area is being added.’

This is significant because it connects the derivative with the integral. This means that differentiation and integration are inverse operations. Derivatives undo integrals.

Question 11

The Fundamental Theorem of Calculus Part 2