AREA
STEPS: to set up a proper definite integral corresponding to a desired area:
1) Make a sketch of the graph of the given function.
2) Shade the region whose area is to be calculated.
3) In choosing the limits of integration, the smaller value of x at ordinate is drawn will be taken as lower limit and the greater as upper limit (i.e., we are to move from left to right on x-axis) and then to evaluate the definite integral.
4) Only the numericals value (and not the algebraic value) of the area will be considered, i.e., we will discard the -ve sign, if some area comes out to be -ve (after calculation).
5) If the curve is symmetrical, then we will find, area of one symmetrical portion and then multiply it by n, if there are n symmetrical portions.
SOME WELL KNOWN CURVES
1) Straight line: ax+ by+ c= 0
2) Circle: x² + y²= a²
3) Parabola: y² = 4ax
4) Ellipse: x²/a² + y²/b²= 1.
5) Hyperbola : x²/a² - y²/b²= 1.
EXERCISE -1
1) Find the area of the triangle formed by the following lines:
i) x= 0, y= 0 and x+ y= 2. 2
ii) x= 0, y= 0 and 3x+ 2y= 6. 3
iii) x= 2, y= 0 and 2x+ y=6. 1
iv) x= 2, y= 0 and y= 3x. 27/8
v) x= 0, y= 1 and 4x+ 3y= 12. 3
2) Find the area of the region bounded by y= 2x+1, the x-axis and the ordinates x=2 and x= 4. 14 sq.units
3) Find the area of the triangle bounded by the x-axis, y-axis and the line x+ y = 4. 8 sq.units
4) Find by Integration the area bounded by the curve y= x², the x-axis and the ordinates x= 1 and x= 3. 26/3 sq.units
5) Find the area region lying in the first quadrant bounded by the parabola y²= 8x, the x-axis and the ordinates x=4. 32√2/3 sq.units
6) Find the area of region bounded by y= 3x² + x , the x-axis and the ordinates x= 1 and x= 3. 30 sq.units
7) Find the area of the plane bounded by the three lines y= x², y= 0 and x= 1. 1/3 sq.units
8) Find the area of region bounded by y²= 4x, x-axis and the straight lines x= 1 and x= 4. 28/3 sq.units.
9) Find the area of the region bounded by y= 4x², y= 0, x= 1 and x= 3. 104/3 sq.units
10) Find the area bounded by the parabola y²= 16x , the x-axis and the ordinates x= 4. 64/3 sq.units
11) Find the area bounded by 4y= x² , and the line x= y. 8/3 sq.units
12) Find the area bounded by the curve y= x(4- x) and x-axis. 32/3 sq.units
13) Find the area of bounded by the parabola y= 16(x-1)(4- x) and the x-axis. 72 sq.units
14) Find the area of bounded by the curve y= 4(x-1)(3 - x) and the x-axis. 16/3 sq.units
15) Find the area of bounded by the curve y= 2x(x-1)(x +1) and the x-axis. 1 sq.units
16) Find the area bounded by y= 3(x -1)(5- x) and x-axis 32 sq.units
17) Find the area of region bounded by y²= 6x and x²= 6y. 12 sq.units
18) Find the area bounded by y²= 4x and x²= 4y. 16/3 sq.units
19) Find the area bounded by y= x² and x= y. 4/3 sq.units
20) Find the area of region bounded by y²= 3x , y-axis and lines y= 1 and y= 4. 7 sq.units
21) Find by integration the area between y²= 8x² + x and the straight line x= 4. 64√2/3 sq.units
22) Find the area enclosed between the axes and the curve (y-2)²= 8x. 1/3 sq.units
23) Draw the graph of y= 3x²+ 2x+4. Shade the area bounded by the curve, x-axis and the lines by x= -1 and x= 3 and hence find its area by integration. 52 sq.units
24) Find the area bounded by the curve y²= 2x² and the line y= x. 2/3 sq.units
25) Find the area included between y²= 9x and x= y. 27/2 sq.units
26) Find the area bounded by the parabola 4y= x² and its latus rectum. 8/3 sq.units
27) Find the area enclosed by the parabola y²= 4ax and its latus rectum. 8a²/3 sq.units
28) Find the area bounded by the parabola y²= 8x and its latus rectum. 32/3 sq.units
29) Find the area bounded by parabola 4y= x² and its latus rectum 8/3 sq.units
30) Find the area of the region bounded by the curve y²= 12x, x-axis and the semi latus rectum. 12 sq.units
31) Find the area cut off from the parabola y²= 12x by its latus rectum. 24 sq.units
32) Find the area above the x-axis bounded by x - 2y +4= 0, x= 3 and x= 6. 51/4 sq.units
33) Find the area of region bounded by the curves y= x² and y= √x. 1/3 sq.units
34) Find the area of the region bounded by y= x² and the line y= 4, (lying in 1st quadrant) 16/3 sq.units
35) Find the area enclosed between the curve y²= x , the x-axis and the ordinates x= 1 and x= 9. 52/7 sq.units
36) Find the area enclosed between the parabola y²= 8 and the straight line x= y. 32/3 sq.units
37) Find the area bounded x-axis, part of the curve by y= (1+ 8/x²) and the ordinates x= 2 and x= 4. If the ordinate at x= a divides the area into equal parts, find a. 4 sq.units, 2√2
38) Calculate the area bounded by the x-axis and the curve y= x - 3√x. 27/2 sq.units
39) Find the area enclosed between the curve by y²= x²(4- x²) , the coordinate axes and the ordinates x=2. 8/3 sq.units
40) Find the area bounded by y= 4x(x-1)(x -2) and the x-axis. 2 sq.units
41) Find the area enclosed by the curve (x-1)(5-x) and the x-axis. 32/3 sq.units
42) Find the area enclosed by the curve y= 3x - x². The x-axis and the ordinates x= 0 and x= 3. 9/2 sq.units
43) Find the area bounded by the curve 4y= x² and the straight line x= 4y -2 27/24 sq.units
44) Find the area enclosed by the curve y= x², the y-axis and the lines y= 1 and y= 9. 52/3 sq.units
45) Find the area of the region lying in the first quadrant bounded by the parabola y²= 4x , the x-axis and the ordinates x= 4. 33/3 sq.units
46) Find the area of region common to the curves y²= 3x and x² = 2y. 4/3 sq.units
47) Shade the area of enclosed by the parabola y²= 8x and x²= y and use the method of integration to find the area so enclosed. 8/3 sq.units
48) Find the area bounded by the curve y= 4x(x -1)(x-2) and the x-axis. 1 sq.units
49) Find the area region common to the curve y= 4x²(3 - x) and the straight line passing through the point (0,0) and (2,16) 1 sq.units
50) Determine the area bounded by the straight line y= 3x+ 3 and the parabolic curve y= x²- 5x +15. 32/3 sq.units
51) Find the area of the region common to the curve y= x² and y= √x. 1/3 sq.units
52) Find the area bounded by the straight lines y= 2x - 4, x = 5 and y= 0. 9 sq.units
53) Find the area enclosed by the parabolic curve y²= 2x +2 and straight line x- y -2= 0 16/3 sq.units
54) Find the area of the circle x²+ y²= a² πa² sq.units
55) Find the area of the ellipse x²/a² + y²/b² = 1. πab sq.units
Continue......
EXERCISE -2
) Showing a rough sketch of the graph y= sin 2x, 0≤ x ≤π/2 find the area enclosed by the curve and x-axis. 1 sq.units
57) Find the area enclosed by the graph y= sin² x, 0≤ x ≤π/2, x-axis and the line x=π/2. Also show a rough sketch of the graph. π/4 sq.units
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