Math 132 Summer 1998

Exam 1

Instructions: Only the right side pages will be graded . Do all scratch work on the left side and then neatly present your solution on the right. Your solution must include not only the answer but also all the work that is necessary to obtain the answer.

1. Find the area of the region bounded by the curves and .

Answer:

> plot( [x,x^2], x=0..1, color = [black,navy]);

> Area := int( x - x^2 , x = 0 .. 1);

2. Find the area of the region bounded by and .

Answer:

> solve( (x/2)^2 = 8 - x , x );

> with(plots):

> implicitplot( {y=x/2,y^2=8-x} , x = - 8 .. 8 , y = -4 .. 2 );

> Area := int( (8-y^2) - 2*y , y = -4 .. 2 );

3. What is the average value of for in the interval ?

Answer:

> Average := 1/(sqrt(3) - 1/sqrt(3))*Int(1/(1+x^2) ,
x = 1/sqrt(3) .. sqrt(3) );

> value(Average);

4. The graph of for in the interval is rotated about the x-axis. What is the volume of the solid of revolution?

Answer:

> implicitplot( y = sqrt(sin(x)) , x = 0 .. Pi , y = 0 .. 1);

> Volume := Int( Pi*sqrt(sin(x))^2 , x = 0 .. Pi );

> value(Volume);

5. The region in the first quadrant that lies under and above is rotated about the y-axis. What is the volume of the solid of revolution?

Answer:

> plot( [4*sqrt(x),(x/2)^3] , x = 0 .. 4 , color = [black,navy] );

Disk-Washer Method:

> solve( y = (x/2)^3 , x );

> solve( y = 4*sqrt(x) , x );

> Volume := Int( Pi*(2*y^(1/3))^2 - Pi*(1/16*y^2)^2 , y = 0 .. 8);

> Volume := value(Volume);

> simplify(Volume);

Cylindrical Shell Method:

> Volume := Int( 2*Pi*x*( 4*sqrt(x)-(x/2)^3 ), x = 0 .. 4 );

> value(Volume);

6. The shaded region in the accompanying figure has a boundary that consists of two line segments and an arc of the circle centered at (0,3). If this figure is rotated about the y-axis, then what is the volume of the resulting solid of revolution ?

Answer:

Cylindrical Shell Method:

> solve( x^2 + (y-3)^2 = 1 , y );

> Volume := Int( 2*Pi*x*( (3+(1-x^2)^(1/2))-3*x ), x = 0 .. 1 );

> value(Volume);

7. The "triangular" region bounded by , , and is rotated about . Express the volume of the solid of revolution by means of an integral. You do not need to evaluate the integral.

Answer:

Disk-Washer Method:

> Volume := Int( Pi*2^2 - Pi*(2-y^(1/3))^2 , y = 0 .. 8 );

> simplify(value(Volume));

Cylindrical Shell Method:

> Volume := Int( 2*Pi*(8-x^3)*(2-x) , x = 0 .. 2);

> value(Volume);

8. A spring is stretched meters beyond its equilibrium position. The number is not noted but it is noted that a force is needed to maintain the spring at that position. It is also noted that of work must be done to stretch the spring an additional (that is, from the unrecorded stretched position). Find the value of the spring constant and determine the value of .

Answer:

> equation1 := 3 = k*xi;

> equation2 := 5 = int(k*x , x = xi .. xi+1);

> solve( { equation1 , equation2 } , { k , xi } );

9. A cable is suspended from the top of a building. The cable is uniform and weighs per foot. A bucket of cement is attached to the end of the cable. The cable is cranked so that the bucket rises . If doing so requires work, then how much does the bucket of cement weigh?

Answer:

Let be the unknown weight of the bucket of cement. We then have the following work equation:

> work_equation := 2000 = W*20 + 10*20 + int( x , x = 0 .. 20);

> solve( work_equation , W );

10. Evaluate . (You must show your work. A calculator-generated numerical answer is not acceptable.)

Answer:

> with(student):

Warning, new definition for D

> J := intparts( Int( x*sin(x) , x = 0 .. Pi ) , x );

> value(J);

11. Evaluate . (You must show your work. A calculator-generated numerical answer is not acceptable.)

Answer:

> with(student):

Warning, new definition for D

> J := Int( ln(x^3) , x = 1 .. exp(1)) ;

> expand(J);

>

>

> J := intparts( 3*Int(ln(x),x = 1 .. exp(1)) , ln(x) );

> value(J);

12. Suppose that is a constant different from . Evaluate .

Answer:

> with(student):

Warning, new definition for D

> J := intparts( Int( x^n*ln(x) , x) , ln(x) );

> value(J);