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APPLICATIONS OF THE DEFINITE INTEGRAL

*      The area between two curves

*      The Volume of the Solid of revolution (by slicing)

 

1. AREA BETWEEN the CURVES

  dx

 

 

 

 

 

 

 

 

 


 

 

EX: Determine the area of the region bounded by  y = 2x2 +10 and y = 4x +16 between x =  – 2  and x = 5

 

Area_Ex4_G1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


EX: Determine the area of the region enclosed by  y = sin x  and  y = cos x   and the y -axis for  

Area_Ex5_G1
 

 

 

 

 

 


EX:  Determine the area of the enclosed area by   and  

 

Area_Ex6_G1Intersection: (-1,-2) and (5,4).

 

 

 

 

 

 

Area_Ex6_G2
 

 

 

 

 

 


  THE SAME: Determine the area of the enclosed area by   and  

Area_Ex6_G3
 

 

 

 

 

 


So, in this last example we’ve seen a case where we could

use either method to find the area. 

However, the second was definitely easier.

 

http://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves_files/eq0054P.gifArea_Ex7_G1EX: Find area

 

 Intersection points are:

 

y = - 1

y = 3

 

Volume of REVOLUTION

▪  Find the Volume of revolution using the disk method

▪  Find the volume of revolution using the washer method

▪  Find the volume of revolution using the shell method

▪  Find the volume of a solid with known cross sections

Area is only one of the applications of integration.  We can add up representative volumes in the same way we add up representative rectangles. When we are measuring volumes of revolution, we can slice representative disks or washers.

 

    DISK METHOD

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

WASHER METHOD

A solid obtained by revolving a region around a line.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Volumes by Cylindrical Shells

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


The Volume for Solids with Known Cross Sections

 


o         Procedure:  volume by slicing   sketch the solid and a typical

o         cross section  find a formula for the area, A(x), of the

o         cross section find limits of integration integrate

o          A(x) to get volume

 

 

 

Find the volume of  a solid whose base is the circle x2 + y2 = 4  and where cross sections perpendicular to the x-axis are

 

a) squares                         

x2 + y2 = 4 y=√(4-x^2 ) length of a side is : 2√(4-x^2 ) dV = A dx A =a^2 V = 4∫_(-2)^2▒〖(4-x^2 )dx =128/3〗
 

 

 

 

 

 

 

 

 

 


b) Equilateral triangles            

x2 + y2 = 4 y=√(4-x^2 ) A= 1/2 a √(a^2-(a/2)^2 )=√3/4 〖 a〗^2=√3 (4-〖 x〗^2 ) V=∫_(-2)^2▒〖√3 (4-〖 x〗^2 )dx=32/√3≈18.475〗
 

 

 

 

 

 

 

 

 

 


x2 + y2 = 4 y=√(4-x^2 ) A= 1/2 π (a/2)^2=1/8 π a^2= π (4-x^2)/2 dV=A dx V=∫_(-2)^2▒〖π (4-x^2)/2 dx〗=16π/3≈16.755  c) semicircles                  

 

 

 

 

 

 

   d) Isosceles right triangles

x2 + y2 = 4 y=√(4-x^2 ) A= 1/2 a (a/2)/tan⁡〖π⁄4〗 =〖 a〗^2/4=4-〖 x〗^2 dV=A dx V=∫_(-2)^2▒〖(4-〖 x〗^2 ) dx=32/3≈10.667〗
 


 

 

 

 

PRACTICE: 

 

1. Find the volume of the solid generated by revolving about the x-axis the region bounded by the graph of        the x-axis, and the line x = 5. Draw a sketch.  1. ANS: 8π       

2. Find the volume of the solid generated by revolving about the x -axis the region bounded by the graph of      where             the x-axis, and the y-axis.  Draw a sketch.   2. ANS: π

3. Find the volume of the solid generated by revolving about the y-axis the region in the first quadrant bounded by the graph of y = x2, the y-axis, and the line y = 6. Draw a sketch.    3. ANS:  18 π                                    

4. Using a calculator, find the volume of the solid generated by revolving about the line y = 8 the region bounded by the graph of y = x 2 + 4, the line y = 8. Draw a sketch.   ANS: 512/15 π

5. Using a calculator, find the volume of the solid generated by revolving about the line y = –3 the region bounded by the graph of y = ex, the y-axis, the lines x = ln 2 and y = – 3. Sketch.  5. ANS: 13.7383 π               

6. Using the Washer, find the volume of the solid generated by revolving the region bounded by y = x 3 and y = x in the first quadrant about the x-axis. Draw a sketch. Method (just a fancy name – use sketch and common sense!!! instead of given boundaries, you have to find it as intersection of two curves and then use sketch to subtract one volume from another )   6. ANS:  4π/21

7. Using the Washer Method and a calculator, find the volume of the solid generated by revolving the region bounded by y = x 3 and y = x about the line y = 2. Draw a sketch.   7. ANS:  17π/21            

8. Using the Washer Method and a calculator, find the volume of the solid generated by revolving the region bounded by y = x2 and x = y2 about the y-axis.  Draw a sketch.   8. ANS: 3π/10

 

 

AGAIN PRACTICE:

1. The base of a solid is the region enclosed by the ellipse    . The cross sections are perpendicular to the x-axis and are isosceles right triangles whose hypotenuses are on the ellipse. Find the volume of the solid.   1. ANS: V = 200/3           

 

 

2. The base of a solid is the region enclosed by a triangle whose vertices are (0, 0), (4, 0) and (0, 2). The cross sections are semicircles perpendicular to the x-axis. Using a calculator, find the volume of the solid.

http://media.wiley.com/Lux/36/39836.nfg008.jpg2. ANS: V = 2.094              

 

3. Find the volume of the solid whose base is the region bounded by the lines x + 4 y = 4, x = 0, and y = 0, if the cross sections taken perpendicular to the x-axis are semicircles.

3. ANS: V = π/6      

 

4. The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y = arctanx, the horizontal line y = 3, and the vertical line x = 1. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?   4. ANS: V = ∫₀¹ (3 - arctan(x))² dx = 6.61233               

 

5. A solid has its base is the region bounded by the lines x + 2y = 6, x = 0 and y = 0 and the cross sections taken perpendicular to x-axis are circles. Find the volume the solid.

5. ANS: 9/2 π              

 

 

6. A solid has its base is the region bounded by the lines x + y = 4, x = 0 and y = 0 and the cross section is perpendicular to the x-axis are equilateral triangles. Find its volume.    6. ANS: V =  16