Fundamental Theorem of Calculus and Integration

How can we express the area under the curve of y = f(t) from t= a to t= x?

A(x) = ∫(x to a) f(t) dt

What is the fundamental theorem of calculus?

F(x) = ∫(x to a) f(t) dt

Which is the antiderivative of f(x) which is to say F’(x) = f(x)

If f(x) and g(x) are continuous functions, and a,b are real numbers:

What does ∫(a(f(x) + bg(x)) dx equal?

a ∫ f(x) dx + b ∫g(x) dx

When do you use integration by parts?

If you have to integrate a product but can’t use integration by substiution

What is the formula for integration by parts?

∫u(x) dv/dx dx = u(x)v(x) - ∫v(x) du/dx dx

Why would you use integration by substitution?

It allows you to integrate functions of functions by simplifying the integral

How do you use integration by substitution?

1. Substitute u for one of the functions of x to give a function that is easier to integrate
2. Find du/dx and rewrite it so that dx is on its own
3. Rewrite the original integral in terms of u and du
4. Integrate as normal

What is the value of the definite integral ∫(b to a) f(x) dx?

F(b) - F(a)

How do you define the rational function f(x)?

A function of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials

What is the t substitution?

t = tan θ/2

What is the effect of the t substitution?

It replaces each trigonometric functions and the differential dθ by algebraic expressions in the new variable t

What is the Arc Length formula of the curve y = y(x) from x=a to x=b?

L = ∫(b to a) √(1 + (dy/dx)^2 dx)

What is the formula for the arc length of a curve defined parametrically eg r(t) = (x(t), y(t), z(t)) where x(t), y(t) and z(t) are all continuously differentiable functions of t?

L = ∫(b to a) √((x’(t))^2 + (y’(t))^2 + (z’(t))^2)