**Unformatted text preview: **MATH-UA.212 Written HW7
Write full, clear solutions to the problems below. It is important that the logic of how you solved these
problems is clear. Although the final answer is important, being able to convey you understand the underlying
concepts is more important. Be sure to write your name and recitation section on your homework, and be
sure to staple all loose pages.
R8
1. (2 points) Find a function f on [3, 8] such that the limit below is equal to 3 f (x) dx.
lim n→∞ n
X
i=1 s 5i
1 + ln 3 +
n 5i · e−3− n · 5
.
n R3√
R3
2. (3 points) Find the exact values of 0 9 − x2 dx and 0 (x − 2) dx by using areas of familiar geometric
shapes. Now use properties of the definite integral to find
Z 3 p
( 9 − x2 + x − 2) dx
0 3. (3 points) Suppose f is even, R −1 f (x) dx = 3, and
R4
there is not enough information to find 1 f (x) dx.
−2 4. Evaluate the following integrals:
Z 1
(a) (2 points)
(x2 + 2)2 dx
0 (b) (2 points) If R1
0 (f (x) − 2g(x)) dx = 6 and R1
0 R4
2 f (x) dx = 5. Either find (2f (x) + 2g(x)) dx = 9, find R1
0 R4
1 f (x) dx, or show (f (x) − g(x)) dx. 5. (2 points) A honeybee population starts with 100 bees and increases at a rate of n0 (t) bees per week.
R 15
What does 100 + 0 n0 (t) dt represent?
6. (2 points) Explain what is wrong with the following calculation of the area under the curve
x = −1 to x = 1:
Z 1
h 1 i1
1 dx
=
− = −1 − 1 = −2.
2
x −1
−1 x 1
x2 from 7. (3 points) Suppose the area under the curve ex from x = 0 to x = a is twenty times the area under the
curve 2e2x from x = 0 to x = b. Solve for a in terms of b. (in other words, write a =(some formula
involving b))
Z 5
1
8. (3 points) Use the Evaluation Theorem to show
dx = ln(5). Now find a fraction which approxi1 x
Z 5
1
mates ln(5), by using M4 (midpoint sum with 4 rectangles) to approximate
dx. (The actual value
x
1
of ln(5) is 1.6094 . . .. For fun, plug your approximation into a calculator and compare.)
9. (2 points) Suppose h is a function such that h(1) = −2, h0 (1) = 3, h00 (1) = 4, h(2) = 6, h0 (2) = 5,
R2
h00 (2) = 13, and h00 is continuous everywhere. Find 1 h00 (u) du. 10. Use the figure below to find the values of
Rb
(a) (2 points) c f (x) dx
Rc
(b) (2 points) a |f (x)| dx 5-2h26
5-2h27
5-2h28
5-2h29 26.
27.
28.
29. Under the curve y = cos x for 0 ≤ x ≤ 2.
Under the curve y = 7 − x2 and above the x-axis.
Above the curve y = x4 − 8 and below the x-axis.
Use Figure ?? to find the values of
(a)
(c) "b
f (x) dx
" ac (b)
(d) f (x) dx a f (x) "c
f (x) dx
"bc
a line segments 5-2h33 |f (x)| dx
5-2h34 Area = 13 ! a b 5-2h29fig "Area = 2 30. Given "0 "2 "2 −2 ! (a) 5-2h30fig −2 Figure 5.37
5-2h31 5-2h31fig "0 x dx and interpret " 2π "1
0 2 e−x dx using n = 5 rectangles to form a
(b) Left-hand sum Right-hand sum 37. (a) On a sketch of y = ln x, represent"the left Riemann
2
sum with n = 2 approximating 1 ln x dx. Write
out the terms in the sum, but do not evaluate it.
(b) On another sketch, represent
" 2the right Riemann sum
with n = 2 approximating 1 ln x dx. Write out the
terms in the sum, but do not evaluate it.
(c) Which sum is an overestimate? Which sum is an underestimate? 5-2h38 38. (a) Draw the rectangles
" π that give the left-hand sum approximation to 0 sin x dx with n = 2. f (x)dx x √ 5-2h37 f (x) 2 cos 36. Estimate 2
−2 0 5-2h36 f (x)dx = 4 and Figure ??, estimate:
−2 f (x)dx
(b)
(a)
0
(c) The total shaded area. "4 35. Without computation, decide if 0 e−x sin x dx is positive or negative. [Hint: Sketch e−x sin x.] Figure 5.36
5-2h30 34. Compute the definite integral
the result in terms of areas. 5-2h35 x c 33. (a) Graph f (x) = x(x + 2)(x − 1).
(b) Find the total area between the graph and the x-axis
between
" 1 x = −2 and x = 1.
(c) Find −2 f (x) dx and interpret it in terms of areas. "0 (b) Repeat part (a) for −π sin x dx.
(c) From your answers to parts (a) and (b), what is
the
" π value of the left-hand sum approximation to
sin x dx with n = 4?
−π 31. (a) Using Figure ??, find −3 f (x) dx.
"
(b) "
If the area of the shaded region is A, estimate
5-2h39 39. (a) Use a calculator or computer to find 6 (x2 + 1) dx.
4
0
f
(x)
dx.
−3
Represent"this value as the area under a curve.
6
(b) Estimate 0 (x2 + 1) dx using a left-hand sum with
1
f (x)
n = 3. Represent this sum graphically on a sketch
4
x
of f (x) = x2 + 1. Is this sum an overestimate or
−4 −3 −2 −1
1
2
3
5
underestimate
" 6 of the true value found in part (a)?
−1
(c) Estimate 0 (x2 +1) dx using a right-hand sum with
n = 3. Represent this sum on your sketch. Is this
Figure 5.38
sum an overestimate or underestimate? ...

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