Question

Use the graph of f(t) = 2t +2 on the interval [−1, 4] to write the function F(x), where
f(x)= integral from 1 to x f(t) dt
f(x)=x^2+3x
f(x)=x^2+2x-12 My Answer
f(x)x^2+2x-3
f(x)=x^2+4x-8

Find the range of the function f(x)=integral from 0 to x (sqrt 4-t^2) dt
(0, 4pi)
(0, 2pi) My Answer
(-4,0)
(0,4)

Answer 1

?[1,x] 2t+2 dt
= t^2+2t [1,x]
= (x^2+2x)-(1^2+2*1)
= x^2+2x-3

?[0,x] ?(4-t^2) dt
= 1/2 ?(4-t^2) + 2 arcsin(t/2) [0,x]
= 1/2 ?(4-x^2) + 2arcsin(x/2) - (1/2 ?(4-0) + 2 arcsin(0))
= 1/2 ?(4-x^2) + 2arcsin(x/2) - 1
The domain is clearly [-2,2]
The range is [-?-1,?-1]

http://www.wolframalpha.com/input/?i=range+1%2F2+%E2%88%9A(4-x%5E2)+%2B+2arcsin(x%2F2)+-+1

Answer 2

Well that was my issue with the second one is I originally got that answer but it isn't an option.