Question

The graph of a function f is shown. Let g be the function that
represents the area under the graph of f between 0 and x.
(a) Use geometry to find a formula for g(x).
(b) Verify that g is an antiderivative of f and explain how this
confirms Part 1 of the Fundamental Theorem of Calculus for
the function f

(Since I can't seem to submit a picture, the graph is f(t)=3t and I tried sketching it below)
| / |
| / |
| / |
| / |
| /_ _ _ _ |_ _ _ _
0 x
How would I solve this?

Answer 1

Oops, the graph got formatted weird after posting, sorry! The far right [ | ] would be where the x is.

Answer 2

if f(x) = 3t, then the area under the graph is just a triangle of base x and height 3x. So its area is g(x) = 1/2 x * 3x = 3/2 x^2

∫[0,x] 3t dt = 3/2 t^2 [0,x] = 3/2 x^2