Integrals Questions

Question

S₁⁴(2x + 3)dx and S₁⁴2xdx + S₁⁴3dx

S₀¹(1-x)dx and S₀¹1dx - S₀¹xdx

They are the same, just one in expanded form.

What is a general result that would apply to S_a^b(f(x) + or - g(x)) dx where f and g are given functions?

Demonstrate graphically (with a particular pair of functions or in general) why the result you stated in the previous question is true.

Answer 1

integral [ f(x) +or- g(x) ] dx, x=a..b

equals

integral f(x) dx, x=a..b +or- integral g(x) dx, x=a..b

that is the LINEARITY property of integrals

Here is a quick proof:

For some partition of x=[a,b],

integral [ f(x) + g(x)] dx , x=a..b

= lim deltaX* [ f(x) +or- g(x) ] as deltaX tends to zer0 <--- definition of integral

= lim deltaX* [ f(x) ] +or- deltaX * [ g(x) ] as deltaX tends to zer0 <-- distributive property

= integral f(x) , x=a..b +or- integral g(x), x=a..b <--- definition of integral