Copy and paste into a word processor and then u can under stand, can anyone please help me out, thanks alot
1. Prove the formula for the sum of the squares of the first n positive integers:
Xn
i=1
i2 = 12 + 22 + 32 + · · · + n2 =
n(n + 1)(2n + 1)
6
2. Find the value of the sum.
(a)
Pn
k=1(k2 + 3k + 4)
(b)
Pn
i=1(i + 1)(i + 2)
(c)
Pn
i=1(3 + 2i)2
3. Find the area under the line y = 2x + 1 from 2 to 5.
4. Find the area under the parabola y = 2x2 + 3 from 0 to 1.
5. Evaluating the following integrals by interpreting each in terms of areas.
(a)
Z 3
0
p9 ? x2 dx (b)
Z 4
2
(2x ? 1) dx (c)
Z 2
?3
(2x + 1) dx
6. Use the properties of integrals and the results
Z b
a
x dx =
b2 ? a2
2
Z
2
0
sinx dx = 1
to evaluate the following integrals. (a)
Z
2
0
(5x ? 2sinx) dx (b)
Z 6
?3
2|x| dx
7. Use the fundamental theorem of calculus to evaluate the following integrals.
(a)
Z 3
0
(x3 ? 5x) dx, (b)
Z 1
0
ex dx, (c)
Z 9
1
2t2 + t2pt ? 1
t2 dt
8. A particle moves along a straight line so that its velocity at time t is v(t) = t2 ? t ? 6 (measured in
meters per second).
(a) Find the displacement of the particle during the time period 1 t 4;
(b) Find the distance travelled during this time period.
9. Find the general indefinite integral.
(a)
Z
xpx dx (b)
Z
px
x2 ?
1
x2
dx (c)
Z
(2 ? px)2 dx
(d)
Z
(cosx ? 2sinx) dx (e)
Z
(2x + secxtanx) dx (f)
Z
x2 + 1 ?
2
x2 + 1
dx
10. The linear density of a rod of length 4 m is given by (x) = 9 + 2px measured in kg per meter,
where x is meaured in meters from one end of the rod. Find the total mass of the rod.
11. Find the derivative of each function using the FTC.
(a) y =
Z x
0
tpt dt (b) y =
Z 3x
0
cost dt (c) y =
Z 3x
2x
u ? 1
u + 1
du
(d) g(x) =
Z x2
sinx
1
p2 + t2
dt (e) h(x) =
Z x3
px
ptsint dt (f) y =
Z 5x
cosx
cos(u2) du
1. Prove the formula for the sum of the squares of the first n positive integers:
Xn
i=1
i2 = 12 + 22 + 32 + · · · + n2 =
n(n + 1)(2n + 1)
6
2. Find the value of the sum.
(a)
Pn
k=1(k2 + 3k + 4)
(b)
Pn
i=1(i + 1)(i + 2)
(c)
Pn
i=1(3 + 2i)2
3. Find the area under the line y = 2x + 1 from 2 to 5.
4. Find the area under the parabola y = 2x2 + 3 from 0 to 1.
5. Evaluating the following integrals by interpreting each in terms of areas.
(a)
Z 3
0
p9 ? x2 dx (b)
Z 4
2
(2x ? 1) dx (c)
Z 2
?3
(2x + 1) dx
6. Use the properties of integrals and the results
Z b
a
x dx =
b2 ? a2
2
Z
2
0
sinx dx = 1
to evaluate the following integrals. (a)
Z
2
0
(5x ? 2sinx) dx (b)
Z 6
?3
2|x| dx
7. Use the fundamental theorem of calculus to evaluate the following integrals.
(a)
Z 3
0
(x3 ? 5x) dx, (b)
Z 1
0
ex dx, (c)
Z 9
1
2t2 + t2pt ? 1
t2 dt
8. A particle moves along a straight line so that its velocity at time t is v(t) = t2 ? t ? 6 (measured in
meters per second).
(a) Find the displacement of the particle during the time period 1 t 4;
(b) Find the distance travelled during this time period.
9. Find the general indefinite integral.
(a)
Z
xpx dx (b)
Z
px
x2 ?
1
x2
dx (c)
Z
(2 ? px)2 dx
(d)
Z
(cosx ? 2sinx) dx (e)
Z
(2x + secxtanx) dx (f)
Z
x2 + 1 ?
2
x2 + 1
dx
10. The linear density of a rod of length 4 m is given by (x) = 9 + 2px measured in kg per meter,
where x is meaured in meters from one end of the rod. Find the total mass of the rod.
11. Find the derivative of each function using the FTC.
(a) y =
Z x
0
tpt dt (b) y =
Z 3x
0
cost dt (c) y =
Z 3x
2x
u ? 1
u + 1
du
(d) g(x) =
Z x2
sinx
1
p2 + t2
dt (e) h(x) =
Z x3
px
ptsint dt (f) y =
Z 5x
cosx
cos(u2) du