Definite Integrals Review
(sections 5.6-5.10)
AP Calculus
Quiz Topics
 Fundamental Theorem of Calculus (section
5.6)
 Nderiv and FnInt on calculator
 Definite integral properties (section 5.7)
 Area using integrals (section 5.8)
Example: Region bounded by
y = -x^2 + 5 and y = -2x + 2
 Volumes of solids (section 5.9)
Fundamental Theorem of Calculus
To evaluate the definite integral of f(x) = 𝑥 2 from -1 to 2:
2 2
𝑥 𝑑𝑥
−1
Let F(x) = 𝑥 2 𝑑𝑥 =
Then
(2)3
3
2 2
𝑥 𝑑𝑥
−1
−
(−1)3
3
=
𝑥3
3
= F(2) – F(-1):
9
3
=3
Integral Properties
 If function values of f(x) are positive and the
interval boundaries are increasing, the integral
will be positive.
 If functional values of f(x) are negative (below x
axis) and the interval boundaries are increasing,
the integral will be negative.
 As a result, it is possible to have areas of positive
and negative area “cancel,” resulting in an integral
of 0.
Odd/Even Integrals with Symmetric Limits
Odd Function: f(-x) = -f(x) Even: f(-x) = f(x)
ODD:
EVEN:
Examples: y = sin x, y = tan x
Examples: y = cos x, 𝑦 = 𝑥 2
𝑦 = 𝑥3
** Interval bounds must be symmetric!!!
Reversal of limits of integration:
𝑏
𝑓
𝑎
𝑥 𝑑𝑥 = −
𝑎
𝑓(𝑥)dx
𝑏
Integral of Constant Times Function:
𝑏
𝑘
𝑎
∙ 𝑓 𝑥 𝑑𝑥 = 𝑘
𝑏
𝑓(𝑥)
𝑎
dx
Integral of Sum:
𝑏
𝑏
𝑓 𝑥 + 𝑔 𝑥 𝑑𝑥 =
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 +
𝑎
𝑔 𝑥 𝑑𝑥
𝑎
Sum of Integrals With Same Integrand
4
5
𝑓 𝑥 𝑑𝑥 +
1
5
𝑓 𝑥 𝑑𝑥 =
4
𝑓 𝑥 𝑑𝑥
1
(Also allows integral to
be broken into more
convenient parts)
Use Common Sense!
2
|
0
𝑥 | dx
8
1 𝑑𝑥
0
Use Common Sense: Not all
Integrals Involve Calculus!
2
|
0
𝑥 | dx
8
1 𝑑𝑥
0
Area Bounded By Curves
 Draw rectangular “strip” between curves and write formula for area
using dx or dy for the width depending on the orientation.
 May need to find intersection points of functions to find interval
boundaries.
Area of “strip” = l x w
= (y1 – y2)dx
Write in terms of x:
Integral is the SUM of strips over interval:
1
4 − 4𝑥 𝑑𝑥 = 2 (𝐴𝑟𝑒𝑎 𝑜𝑓 𝑟𝑒𝑔𝑖𝑜𝑛)
0
Area bounded by curves: “Sideways”
 Functions on left and right instead of top and bottom. Make
sure “strip” always extends from one function to another
(Not one function back to itself)
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑟𝑒𝑔𝑖𝑜𝑛:
𝑑
𝑓 𝑦 − 𝑔 𝑦 𝑑𝑦
𝑐
Integral Vs. Area
 Be able to write the INTEGRAL EXPRESSION for the area.
Additional Topics
 Make sure you can find basic integrals (look over trig
derivatives list – pay attention to SIGNS!)
 Chain Rule for integrals: (3𝑥 + 5)6 𝑑𝑥
 “Tricky” integrals – change form, make graphs, etc:


𝑥 𝑑𝑥
1
𝑑𝑥
3
𝑥
𝑥 − 2 𝑥 + 1 𝑑𝑥
2𝑥(4𝑥 2 − 3)5 𝑑𝑥
1
𝑑𝑥
(2𝑥−5)2
2𝑥 − 1𝑑𝑥
 Look out for odd/even functions (can graph to check)
 Draw original function given f’(x)
Additional Topics
 Know how to find functional integral (fnInt) on graphing




calculator
Be able to write equations for simple volume of solids (disc
method)
Cylinder Volume: 𝜋 𝑟 2 ℎ
Use trapezoidal rule/Riemann sums to estimate area using a table
of values (no function given). (PLOT GRAPH!!!!)
Application problems: Area under curve for velocity-time
graph is displacement!
 Review Problems: pg. 261: R6 bcd, R7, R8, R10 (trap rule on
part c)