SME
Calculus I · Problem type
The area function and the Fundamental Theorem, part 1
Use this for “F(x) is the integral of 3t − 2 from 1 to x. Find F(4) and F′(4).”
Chart
f(3) = 7
F(x) = ∫ from 0 to x of f(t) dt: F(3) = 12, and its slope there is f(3) = 7
This is an example. Type your own values to start a new problem.
F(
x
)
=
∫
(
m
t
+
c
)
dt
=
F(x)
F′(
x
)
=
F′(x)
Formulas
F(
x
)
=
m
×
x
2
−
a
2
2
+
c
×
(
x
−
a
)
12
=
2
×
3
2
−
0
2
2
+
1
×
(3
−
0)
F′(
x
)
=
m
×
x
+
c
7 = 2 × 3 + 1
Assumptions
•
F(x) = ∫ from a to x of f(t) dt collects the signed area under f(t) = mt + c from a up to x. The letter t runs along the axis; x is where it stops.
•
An antiderivative of mt + c is G(t) = mt² ÷ 2 + ct, so F(x) = G(x) − G(a), by the Fundamental Theorem (part 2).
•
The Fundamental Theorem (part 1): F′(x) = f(x). Moving x on adds a thin strip as tall as f(x), so the area grows at that rate.
•
Area below the t-axis counts as negative, and F(a) = 0. When x is left of a, F(x) is minus the area from x to a.
Step-by-step
WE KNOW
m = 2, c = 1, a = 0, x = 3
FIND
area function at x (F(x)), slope of the area function at x (F′(x))
Step 1 · Find area function at x (F(x))
Use F(x) = m × (x² − a²) ÷ 2 + c × (x − a)
An
antiderivative
of
f
(t)
=
2t
+
1
is
G(t)
=
t
2
+
t.
Take
G
at
x
less
G
at
a
:
m
(
x
2
−
a
2
)
2
+
c
(
x
−
a
).
F(
x
)
=
2
×
3
2
−
0
2
2
+
1
×
(3
−
0)
3
∫
0
(2
t
+ 1) dt
=
[
t
2
+
t]
from
0
to
3
=
12
F(
x
)
=
12
Step 2 · Find slope of the area function at x (F′(x))
Use F′(x) = m × x + c
By
the
Fundamental
Theorem
(part
1),
the
slope
of
F
at
x
is
the
height
of
f
at
x
:
put
t
=
x
into
mt
+
c
.
No
integral
is
needed.
F′(
x
)
=
2
×
3
+
1
F′(
x
)
=
6
+
1
F′(
x
)
=
7
Check
Put the numbers back in. Both sides should match.
12
=
2
×
3
2
−
0
2
2
+
1
×
(3
−
0)
✓
7 = 2 × 3 + 1 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Definite integrals and the Fundamental Theorem
Main lesson
Riemann sums
Use this for “Estimate the integral of x² from 1 to 4 with 6 right-endpoint rectangles, then find the error.”
Average value of a function
Use this for “Find the average value of f(x) = 2x² on [1, 4], and the c in [1, 4] where f(c) equals it.”
Displacement and distance from velocity
Use this for “A particle moves with v(t) = 3t − 6 m/s for 0 ≤ t ≤ 4 s. Find its displacement and the total distance it travels.”
Area between a line and a parabola
Use this for “Find the area of the region between y = 2x + 3 and y = x².”