Math 220 (section AC2)
Practice Exam 4
Spring 2012
Name
• Do not open this test until I say start.
• Turn off all electronic devices and put away all items except a pen/pencil and an eraser.
• No calculators allowed.
• You must show sufficient work to justify each answer.
• Quit working and close this test booklet when I say stop.
Expectations:
section 5.3-5.5
• be able to precisely state both parts of the fundamental theorem of calculus.
• be able to decide when functions satisfy the conditions of this theorem, and be able to
evaluate definite integrals with this theorem when they do.
1
√ 1
• Know indefinite integral formulas for 0, k, sin x, cos x, sec2 x, csc2 x, sec x tan x, csc x cot x, ex , 1+x
2,
1−
• Know that the definite integral of a rate of change gives the total change. Be able to use
this net change theorem for applied problems involving rates of change such as velocity
and acceleration.
• be able to solve a wide variety of definite or indefinite integrals using substitution.
sections 6.1 Be able to find area between two curves. (This may require breaking area up
into sum of two or more integrals.)
section 6.2-6.3
• Be able to find volumes for solids formed by revolving a region around any vertical or
horizontal line. Be able to recognize when cylindrical shells are necessary.
• Be able to find volumes for solids formed by building upon some base and having crosssections which are rectangles, squares, triangles, semi-circles, etc.
• be able to integrate with respect to x or y to determine these volumes.
section 6.4
• Be able to find work required to move a spring.
• Be able to find work required to pump water out of a tank (of various shapes.)
• Be able to find work required to pull a hanging chain up.
section 6.5
• Be able to find the average value of a function.
• Know the graphical interpretation of the average value of a function.
• Know the mean value theorem for integrals.
1. Find the following integrals
(a)
(b)
(c)
(d)
(e)
(f)
Z
2ex + sec x tan x dx
Z
x5 − 3x
dx
x2
Z
√
x + 4x5 + xπ dx
Z
cos x − (1 − x2 )−1/2 dx
Z
Z 2
1
(g)
(h)
(i)
(j)
(k)
(l)
3 sin r − 2r dr
24x
dx
1 + 4x2
Z
3x8
x3 + 4
Z
1
dx
x ln x
Z
√
Z
ee ex dx
Z
Z
e2x
dx
ex + 1
x
ln(cos x) tan x dx
√
x
dx
1 − x4
2. Find the derivative of
Z e x2
3
1−t
dt.
1 + t2
3. Find the average value of the function f (x) =
4. Find the derivative
(a)
Z 0
x3
(b) (
Z x
1
dt)3
1 + sin2 t
Z sin x
a
(d) sin(
(e)
x
1
dt
1 + sin2 t
Z x
a
Z b
of
1
dt
1 + sin2 t
a
(c)
d
dx
1
dt)
1 + sin2 t
1
dt
1 + + sin t
t2
√ x
x2 +9
on the interval [1, 4].
5. work problems from the 6.4 homework, like conical tanks or spring problems.
6. Let R be the finite region bounded by the graph of f (x) = 5x − x2 and the x-axis on the
interval [0, 5]. Set up, but do not evaluate, definite integrals which represent the given
quantities.
(a) The average value of f on the interval [0, 5]
(b) the area of R
(c) The colume of the solid obtained when R is revolved around the horizontal line y =
−10
(d) The volume of the solid obtained when R is revolved around the vertical line x = 8.
7. Let R be the finite region bounded by the graph of f (x) = 4x − x2 and y = x.
(a) Find the volume of the solid obtained by rotating R around the y-axis.
(b) Find the volume of the solid obtained by rotating R around the line y = −1.
(c) Find the area of R.
8. Let R be the finite region bounded by y = e2x , y = 3 and the y-axis. Set up but do
not evaluate the volume of the solid obtained when R is revolved around the y-axis in the
following manner:
(a) Integrate with respect to x
(b) Integrate with respect to y
9. Find the volume of the doughnut, obtained by rotating the circle (x − a)a + y 2 = b2 (a > b)
about the vertical axis.
10. FInd the volume of the ”infinite trumpet” obtained by rotating the region between the
x-axis and the function f (x) = 1/x, x ≥ 1 about the x-axis.
11. Find the area between tan x and cos x on the interval [0, 1].
12. Evaluate:
Z 5
−4
(x +
√
52
−
x2 ) dx
+
Z −4 √
−5
52 − x2 dx