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Group Work 1, Section 5.
The Area Function
Recall that we can use the notation
∫ (^) b a f^ (t)^ dt^ to denote the area under the curve^ f^ (t)^ between^ t^ =^ a^ and t = b.
- Consider the constant function f (t) = 4. y
1 t
4
x
(a) Using geometry, compute
1 f^ (t)^ dt.
(b) Similarly compute
1 f^ (t)^ dt^ and^
1 f^ (t)^ dt.
(c) Using your answers to parts (a) and (b) as a guide, compute
∫ (^) x 1 f^ (t)^ dt^ for any^ x^ ≥^1.
(d) We now define the area function A (x) =
∫ (^) x 1 f^ (t)^ dt,^1 ≤^ x^ ≤^4. What is^ A^ (2)?^ A^ (2.5)?^ A^ (1)? Write a general formula for A (x).
The Area Function
- Let f (t) = 2t + 2 for all t.
(a) Using geometry, compute
0 f^ (t)^ dt.
(b) Similarly, compute
0 f^ (t)^ dt.
(c) Using your answers to parts (a) and (b) as a guide, compute
∫ (^) x 0 f^ (t)^ dt^ for any^ x^ ≥^0.
(d) We now define another area function B (x) =
∫ (^) x 0 f^ (t)^ dt. What is^ B^ (2)?^ B^ (4)?^ B^ (0)? Write a general formula for B (x).
(e) We will now define a third area function C (x) =
∫ (^) x − 1 f^ (t)^ dt^ for any^ x^ ≥ −^1 , as pictured below:
What is C (2)? C (4)? C (−1)? Write a general formula for C (x).
