Math 21B Final
1) Which two mathematicians are said to have invented calculus?
(2 points)
2) What is a "mesh"?
What "happens" to it when we find a definite integral?
Why?
(6 points)
3) State the Mean Value Theorem for Definite Integrals.
(6 points)
4) What is an "elementary" function?
What kinds of functions are not elementary?
(4 points)
5) Find
x
2
e
4
x
dx
∫
(6 points)
6) Find
4
x
2
3
+
8
x
3
dx
∫
(6 points)
7) Let
F x
( 29 =
e

t
2
dt
0
x
∫
.
Is
F
(
x
) concave up or concave down at
x
= 2?
Justify.
(4 points)
8) Find the centroid of this triangle by following these four steps.
a) Find equations for the two diagonal lines.
(4 points)
b) Find the moment about the
y
axis by starting with integrals like
(6 points)
...
0
b
∫
+
...
b
a
∫
c) Find the area of the triangle.
(4 points)
d) Find
x
.
(4 points)
9) Find
1
9
+
x
2
dx
∫
.
(15 points)
Here are some integrals that might be of assistance:
sec
xdx
∫
=
lnsec
x
+
tan
x
sec
2
xdx
∫
=
tan
x
sec
3
xdx
∫
=
1
2
sec
x
tan
x
+
1
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 Spring '08
 Vershynin
 Calculus, Derivative, tan xdx, sec xdx

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