*Further Integration Flashcards
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
π∫ (y)^2 dx
- Square function
- Integrate this new function
- Substitute definite values
- 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
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
π∫ (x)^2 dy
- Make x subject (if you have to)
- Square function
- Integrate this new function
- Substitute definite values
- Multiply by π
E.g. volume of y=x^2 from y (0 to 4) = π((y^2)/2) [0,4]
= 8π units^3
How do you use u substitution?
E.g for ∫ x√(x-1) dx with u = x - 1
- Derive u
- Find equivalence in original equation
- Alter equation into y = ∫u-equivalent du
- If discrete, sub x-values into u-equation, If not add constant C
E.g. ∫ x√(x-1) dx = ∫ (u-1)√(u) du
What is a form of cos^2(nx) that can be integrated?
(½)(1 + cos(2nx))
What is a form of sin^2(nx) that can be integrated?
(½)(1 - cos(2nx))
How do you integrate cos^2(nx)?
E.g. ∫ 5cos^2(3x) dx
- Change to form that can be integrated
- Integrate normally
- Add constant C
E.g. ∫ 5cos^2(3x) dx = 5∫ (½)(1 + cos(6x)) =
5/2(x + sin(6x)/6) + C
How do you integrate sin^2(nx)?
E.g. ∫ 2sin^2(2x) dx
- Change to form that can be integrated
- Integrate normally
- Add constant C
E.g. ∫ 2sin^2(2x) dx = 2∫ (½)(1 - cos(4x)) =
x - sin(4x)/4) + C
How do you integrate dx/(√(a^2 - x^2))?
E.g. ∫ 1/(√(4 - 25x^2)) dx
- Factorise into a form where it’s just (x^2)
- With (x^2) form change, find square root of value next to (x^2) (this is a)
- Form sin^-1(x/a) (times whatever is necessary)
- Add constant C
E.g. ∫ dx/(√(4 - 25x^2)) dx = (1/5)sin^-1(5x/2) + C
How do you integrate -(dx/(√(a^2 - x^2)))?
E.g. ∫ -(dx/(√(16 - x^2))) dx
- Factorise into a form where it’s just (x^2)
- With (x^2) form change, find square root of the value next to the (x^2) (this is a)
- Form cos^-1(x/a) (times whatever is necessary)
- Add constant C
E.g. ∫ -(dx/(√(16 - x^2))) dx = cos^-1(x/4) + C
How do you integrate dx/(a^2 + x^2)?
E.g. ∫ dx/(81 + 9x^2) dx
- Factorise into a form where it’s just (x^2)
- With (x^2) form change, find square root of value next to (x^2) (this is a)
- Form tan^-1(x/a) (times whatever is necessary)
- Divide by a also
- Add constant C
E.g. ∫ dx/(81 + 9x^2) dx = (1/27)tan^(x/3) + C