4.4 Fundamental Theorem of Calculus 3
Recall yesterday when we evaluated ∫ (x + 2)dx
0
We found the area under that curve both geometrically (area of a trapezoid) and with the calculator. Today we will actually find the area using calculus. Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F(x) is the ________________________ of f(x) on that interval, then b
∫ f (x)dx = _________________________=_______________________________ a
*** Evaluate with the calculator first as we did yesterday 3
∫ (x + 2)dx =
0
Now evaluate using calculus 2
∫ (x
2
− 3)dx
1
4
∫3
1
x dx
∫
2
0
(2x 2 − 3x + 2)dx
∫
−1
−2
x−
1
x2
When an object is moving vertically we know that its acceleration is -­‐32 feet per second per second, a(t) = −32 . If a ball is thrown vertically upward from the ground with an initial velocity of 60 feet per second, it is thrown from a height of 6 feet. What function would describe its velocity? What function would describe its position in relation to the ground?