Thomas Calculus 13th [Solutions]

Section 5.4 The Fundamental Theorem of Calculus 369
Sums of rectangles evaluated at midpoints can be represented and evaluated by this set of commands.
Clear[x, f, a, b, n]
{a, b} {0, }; n 10; dx (b a)/n;
2
f Sin[x] ;
xvals Table[N[x], {x, a dx/2, b dx/2, dx}];
yvals f /.x xvals;
boxes MapThread[Line[ {{# 1, 0},{ # 1, # 3},{ # 2, # 3},{ # 2, 0}] &,{xvals, dx/2, xvals dx/2, yvals}];
Plot[f, {x, a, b},Epilog boxes];
Sum[yvals[[i]] dx, {i, 1, Length[yvals]}]//N
5.4 THE FUNDAMENTAL THEOREM OF CALCULUS
1.
2 2 3 2 3 2
2 3 (2) 3(2) (0) 3(0)
x( x 3) dx ( x 3 x) dx x x
10
0 0 3 2
0
3 2 3 2 3
3 2 2
2.
1 3 3
2 2 (1) 2 ( 1) 2
x 2x 3 dx x x 3 x
(1) 3(1) ( 1) 3( 1) 20
1 3
1
3 3 3
3 1
3.
2
2
3 1 1 1 1 124
dx
1
4 3 3 3
2 ( x 3) ( x 3) 125 125
2 (5) (1)
4.
300
1
1 299 x 1 300 300 1
x dx
(1) ( 1) 1 1 0
1 300 300 300
1
5.
4
3 4
4
4 4
2 x
3 x
3 4 3 1 1 753
3x dx x
4 1 64 16 1
4 16 16 16 16 16
1 1
6.
3
4
3 x 2
x 2x 3 dx x 3x
2
4
4 4
3 2 ( 2) 2 81 105
3 3(3) ( 2) 3( 2) 6
4 4 4 4
4
1
7.
1 2 2 3/2
x x dx x x 1 2
0 3 3
0
3 3
3 1
0 1
8.
32 6/5 1/5
32
x dx 5 x
5 ( 5) 5
1 1 2 2
9.
/3 2 /3
2sec x dx [2 tan x]
0
0 2 tan (2 tan 0) 2 3 0 2 3
3
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