Question

Evaluate the following:

`int_1^4` f(x) dx where f(x) = `{{:(4x + 3",", 1 ≤ x ≤ 2),(3x + 5",", 2 < x ≤ 4):}`

Answer 1

`int_1^4` f(x) dx = `int_1^2 "f"(x)  "d"x + int_2^4 "f"(x)  "d"x`

= `int_1^2 (4x + 3)  "d"x + int_2^4 (3x + 5)  "d"x`

= `[4(x^2/2) + 3x]_1^2 + [3(x^2/2) + 5x]_2^4`

= `[2x^2 +3x]_1^2 + [3/2 x^2 + 5x]_2^4`

= `[2(2)^2 + 3(2)] - [2(1)^2 + 3(1)] + [3/2(4)^2 + 5(4)] - [3/2 (2)^2+ 5(2)]`

= `[2(4) + 6] - [2 + 3] + [3/ (16) + 20] - [3/2 (4) + 10]`

= [8 + 6] – [5] + [24 + 20] – [6 + 10]

= 14 – 5 + 44 – 16

= 58 – 21

= 37