*Further Integration

Question 1

How do you find the volume of a solid (bound by x-axis)?

E.g. volume of function y=x^2 bound by x-axis from x = 0 to 2

Answer 1

π∫ (y)^2 dx

1. Square function
2. Integrate this new function
3. Substitute definite values
4. Multiply by π
   E.g. volume of y=x^2 from x (0 to 2) = π(x)^5/5 [0,2]
   = 32π/5 units^3

Question 2

How do you find the volume of a solid (bound by y-axis)?

E.g. volume of function y=x^2 bound by x-axis from y = 0 to 4

Answer 2

π∫ (x)^2 dy

1. Make x subject (if you have to)
2. Square function
3. Integrate this new function
4. Substitute definite values
5. Multiply by π
   E.g. volume of y=x^2 from y (0 to 4) = π((y^2)/2) [0,4]
   = 8π units^3

Question 3

How do you use u substitution?

E.g for ∫ x√(x-1) dx with u = x - 1

Answer 3

1. Derive u
2. Find equivalence in original equation
3. Alter equation into y = ∫u-equivalent du
4. If discrete, sub x-values into u-equation, If not add constant C
   E.g. ∫ x√(x-1) dx = ∫ (u-1)√(u) du

Question 4

What is a form of cos^2(nx) that can be integrated?

Answer 4

(½)(1 + cos(2nx))

Question 5

What is a form of sin^2(nx) that can be integrated?

Answer 5

(½)(1 - cos(2nx))

Question 6

How do you integrate cos^2(nx)?

E.g. ∫ 5cos^2(3x) dx

Answer 6

1. Change to form that can be integrated
2. Integrate normally
3. Add constant C
   E.g. ∫ 5cos^2(3x) dx = 5∫ (½)(1 + cos(6x)) =
   5/2(x + sin(6x)/6) + C

Question 7

How do you integrate sin^2(nx)?

E.g. ∫ 2sin^2(2x) dx

Answer 7

1. Change to form that can be integrated
2. Integrate normally
3. Add constant C
   E.g. ∫ 2sin^2(2x) dx = 2∫ (½)(1 - cos(4x)) =
   x - sin(4x)/4) + C

Question 8

How do you integrate dx/(√(a^2 - x^2))?

E.g. ∫ 1/(√(4 - 25x^2)) dx

Answer 8

1. Factorise into a form where it’s just (x^2)
2. With (x^2) form change, find square root of value next to (x^2) (this is a)
3. Form sin^-1(x/a) (times whatever is necessary)
4. Add constant C
   E.g. ∫ dx/(√(4 - 25x^2)) dx = (1/5)sin^-1(5x/2) + C

Question 9

How do you integrate -(dx/(√(a^2 - x^2)))?

E.g. ∫ -(dx/(√(16 - x^2))) dx

Answer 9

1. Factorise into a form where it’s just (x^2)
2. With (x^2) form change, find square root of the value next to the (x^2) (this is a)
3. Form cos^-1(x/a) (times whatever is necessary)
4. Add constant C
   E.g. ∫ -(dx/(√(16 - x^2))) dx = cos^-1(x/4) + C

Question 10

How do you integrate dx/(a^2 + x^2)?

E.g. ∫ dx/(81 + 9x^2) dx

Answer 10

1. Factorise into a form where it’s just (x^2)
2. With (x^2) form change, find square root of value next to (x^2) (this is a)
3. Form tan^-1(x/a) (times whatever is necessary)
4. Divide by a also
5. Add constant C
   E.g. ∫ dx/(81 + 9x^2) dx = (1/27)tan^(x/3) + C