Integration - Topic 8

Question 1

Year 1 - Chapter 13.1

What is the Second Fundamental Theorem of Calculus?

Answer 1

If dy/dx = kxⁿ, then
y = (k/(n+1))xⁿ⁺¹ + c
n ≠ 1

dy/dx = f’(x)

Question 2

Year 1 - Chapter 13.1

Integrate:

1. dy/dx = (1/2)x²
2. f’(x) = -3x⁻¹/²
3. f’(x) = keᵏˣ

Answer 2

1. y = (1/6)x³
2. f(x) = -6x¹/²
3. f(x) = eᵏˣ

Question 3

Year - Chapter 13.4

Calculate the definite integral:
²∫₁ 3x² dx

Answer 3

1. [x³]²₁
2. (2³) - (1³)
3. 8-1
4. ²∫₁ 3x² dx = 7

Question 4

Year 1 - Chapter 13.5

How do you calculate the area under a curve?

When the curve is above the horizontal x-axis

Answer 4

Area = ᵇ∫ₐ y dx, where y = f(x)

Question 5

Year 1 - Chapter 13.6

How do you calculate the area under a curve?

When the curve is above and below the horizontal x-axis

Answer 5

Area = ᵇ∫₀ y dx - ⁰∫ₐ y dx, where y = f(x)

Question 6

Year 1 - Chapter 13.7

How do you calculate the area bound by a curve and a linear equation?

Answer 6

1. Equate y = f(x) to y = x
2. Find the roots
3. Find the area beneath the curve; ᵇ∫ₐ y dx
4. Find the area beneath the triangle; 1/2 x base x height
5. Area = area beneath the curve - area beneath the triangle

Question 7

Year 1 - Chapter 13.7

Calculate the area bound by the line and curve:
y = x(4-x), y = x

Answer 7

1. x(4-x) = x
2. 4x-x² = x
3. 3x-x² = 0
4. x(3-x) = 0
5. x = 0, or 3
6. ³∫₀ x(4-x) dx = [2x²-(x³)/3]³₀ = 9
7. 1/2 x 3 x 3 = 4.5
8. 9 - 4.5 = 4.5 = Area

Question 8

Year 1 - Chapter 13.2

Calculate the indefinite integral:
∫ xⁿ dx

Answer 8

(xⁿ⁺¹)/(n+1) + c

Question 9

Year 2 - Chapter 11.1

What are the 9 standard integrating functions?

Answer 9

1.
2.
3.
4.
5.
6.
7.
8.
9.

Question 10

How do you use the reverse chain rule?

Question 11

How do you use the reverse product rule?

Question 12

How do you use integration by substitution?

Question 13

How do you use integration by parts?