Calculus: Analytic Geometry and Calculus, with Vectors

(^230) Integrals
opinion. Look at Figure 4.392 and note that S seems to fill up about one-third
of the square having opposite vertices at the points (0,0) and (1,1), and hence
that the area ISI of S should be about one-third. Now comes the calculation.
Show that
f 1x2 dz = g. 1
0
5 Sketch a graph of the equation y x + 1 over the interval 1 5 x 5 3
and use elementary geometrical ideas to find the area of the part of the plane
bounded by this graph and the graphs of the equations x = 1, x = 3, and y = 0.
Then evaluate the integral

f13(x+1)dx

and find out whether we obtain more experimental verification of the cheerful

opinion of the preceding problem.

6 Figures 4.393 and 4.394 show graphs of y = sin x and y = cos x over the

interval 0 S x _< v. Observe that a* particular region of Figure 4.393 seems to

Y

Figure 4.393

1

0

r

x

-1

Figure 4.394

fill about two-thirds of the enclosing rectangle and hence that the region ought

to have area about 2ir/3. Then obtain the first of the formulas

or /2

fo sinx dx = 2,

for

cos x dx = 1,

f"7/2

a/2

cos x dx = -1

and give an interpretation of the result. Then obtain the second and third

formulas and interpret the results in terms of regions in Figure 4.394.

7 Prove that if u and v have continuous derivatives over the interval

a<x5b,then

fab

v(x) u'(x) dx.

fab

u(x)v'(x) dx = u(x)v(x) I.

Hint: Decide how the formula

f ab F(x) dx= F(x)]a

can be used. Remark: The formula to be proved is one of the most useful

formulas in the calculus; it is the formula for integration by parts.

8 Some of the most important applications of integrals involve inequalities,