Calculus: Analytic Geometry and Calculus, with Vectors

240 Integrals

5 Use the technique of the text to find the area of the triangular patch

bounded by the lines having the equations y = 2x, y = 0, and x = 3. Check

your answer by use of elementary geometry.

6 Let f4 be the area of the region bounded by the x axis and the graph of

the equation y = x(1 - x). Sketch an appropriate graph showing a sample

rectangle and fill in the details involving the formula

A = lim x(1 - x) Ox = 101 (x - x22) dx = 9.

Find the area of the region bounded by the coordinate axes and the graph

of the equation y = x3 - 8. 14ns.: 12.

8 Find the area of the part of the plane bounded by the graphs of the equa-

tions y = x3 - 3x and y = x. 11ns.: 8.

9 Find the area of the region bounded by the graphs of the equations y = x,

y = 2x, and y = x2. 11ns.:

10 Find the area of the region in the first quadrant bounded by the x axis

and the graphs of the equations y = x and y = 2 - x2. 14ns.: (8 ' - 7)/6.

11 Let 4 be the area of the part of the plane which lies between the lines

having the equations x = 9r and x = 21r and is bounded by the x axis and the

graph of the equation y = sin x. Sketch an appropriate graph showing a sample

rectangle and, observing that the height of the rectangle is the positive number

• sin x (not the negative number sin x), fill in the details involving the formula
  r
  '4 = lim (- sin x) Ax

2A

= - r sin x dx = 2.

12 Someday we will be able to show that the graph of the equation x3 + yil

= a is a positive constant, a part of a parabola. Find the area of the

region bounded by the graph and the coordinate axes. Ans.: a2/6.

13 Is the area of the region bounded by the graphs of the equations

y=x3+x2, Y=x3+1

the same as the area of the region bounded by the graphs of the equations

Y =x2, Y = 1?

14 The graph of each of the following equations contains a loop; determine

the nature of the graph and find the area of the region bounded by the loop, it

being assumed that a is a positive constant.

(a) y2 = x(a - x)2 Ans.: Aa,56

(b) y2 = x(x - a)2 .Ins.: Aa%

15 The graphs of the equations y = jx2 and y = x + 4 bound a region R.

With the aid of a reasonably good figure, make an estimate of the area JRI of R.

Then find JRf by making partitions of an appropriate part of the x axis so that

vertical strips appear in the calculation. Then find JRI by a method in which

horizontal strips appear. Make the results agree with each other and use your

estimate to provide assurance that the two answers are reasonable.