SME
Calculus I
Definite integrals and the Fundamental Theorem
Use this for “Evaluate the integral of 3x² − 4x + 2 from 0 to 2.”
Chart
Shaded: ∫ from 1 to 3 of f(x) dx = 32/3
All of it is above the axis, so the integral is the shaded area
This is an example. Type your own values to start a new problem.
∫
(
p
x²
+
q
x
+
r
)
dx
=
32
3
I
F(a)
Antiderivative at the lower limit
worked out (not typed)
F(b)
Antiderivative at the upper limit
worked out (not typed)
Formulas
F(
a
)
=
p
×
a
3
3
+
q
×
a
2
2
+
r
×
a
4
3
=
1
×
1
3
3
+
0
×
1
2
2
+
1
×
1
F(
b
)
=
p
×
b
3
3
+
q
×
b
2
2
+
r
×
b
12
=
1
×
3
3
3
+
0
×
3
2
2
+
1
×
3
I
=
F(
b
)
−
F(
a
)
32
3
=
12
−
4
3
Assumptions
•
An antiderivative of f(x) = px² + qx + r is F(x) = px³ ÷ 3 + qx² ÷ 2 + rx: by the power rule, F′(x) = f(x).
•
The Fundamental Theorem (part 2): the integral of f from a to b is F(b) − F(a). A + C would cancel, so it is left out.
•
Area below the x-axis counts as negative, so the integral is the area above the axis less the area below it.
•
Swapping the limits flips the sign: the integral from b to a is −I.
Step-by-step
WE KNOW
p = 1, q = 0, r = 1, a = 1, b = 3
FIND
antiderivative at the lower limit (F(a)), antiderivative at the upper limit (F(b)), value of the integral (I)
Step 1 · Find antiderivative at the lower limit (F(a))
Use F(a) = p × a³ ÷ 3 + q × a² ÷ 2 + r × a
Raise
each
power
by
1
and
divide
by
the
new
power:
F(x)
=
x
3
3
+
x.
Put
in
x
=
a
.
F(
a
)
=
1
3
3
+
1
F(
a
)
=
4
3
Step 2 · Find antiderivative at the upper limit (F(b))
Use F(b) = p × b³ ÷ 3 + q × b² ÷ 2 + r × b
The
same
antiderivative,
at
x
=
b
.
F(
b
)
=
3
3
3
+
3
F(
b
)
=
12
Step 3 · Find value of the integral (I)
Use I = F(b) − F(a)
Subtract
F(
a
)
from
F(
b
):
the
change
in
the
antiderivative
from
a
to
b
.
I
=
12
−
4
3
3
∫
1
(
x
2
+ 1) dx
=
[
x
3
3
+
x]
from
1
to
3
=
32
3
I
=
32
3
Check
Put the numbers back in. Both sides should match.
4
3
=
1
×
1
3
3
+
0
×
1
2
2
+
1
×
1
✓
12
=
1
×
3
3
3
+
0
×
3
2
2
+
1
×
3
✓
32
3
=
12
−
4
3
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Riemann sums
Use this for “Estimate the integral of x² from 1 to 4 with 6 right-endpoint rectangles, then find the error.”
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).”
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.”