Integrals Problem

Question

Find ∫₀⁵ f(x)dx if

f(x)={3 for x<3

{x for x≥3.

Answer 1

integral from 0 to 5 = integral from x = 0 to 3 plus integral from x= 3 to 5

integral from 0 to 3 is 3x = 3(3) = 9

integral from 3 to 5 is (1/2)x^2 = (5^2 - 3^2)/2 = (25-9)/2/2 = 16/2 = 8

Integral = 8

add 9+8 to get integral from 0 to 5 = 17

the integral is the area under the curve

or the area under y=3 from x=0 to x=3, which is 3 times 3 = 9

plus the area under y=x from x =3 to x =5, that is the area of a rectangle 2 by 3 = 6 plus the area of a triangle 2 by 2 = 4/2 = 2. 9+6 + 2 = 17