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# Areas of bounded regions

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### Areas of bounded regions

1. 1. - AREAS OF BOUNDED REGIONS Himani Asija Delhi Public School Vasant Kunj
2. 2. THEOREM : Let f(x) be a•continuous function defined in [a,b].Then the area bounded by the curve , the x axis and the ordinates a and bis given by b b ∫ f ( x)dx = ∫ ydx a a or b b ∫ f ( y )dy = ∫ xdy a a
3. 3. With different values of n, let’s see what the area looks like ! def int 3
4. 4. Types of integrals to be evaluated :y y=f(x) Type 1 region : S: Bounded by two curves y=f(x) and g(x) and between two vertical line y=a and y=b. So a b x S = {( x, y ) | a ≤ x ≤ b, g ( x) ≤ y ≤ f ( x)} b y=g(x) A = ∫ [ f ( x) + {− g ( x)}]dx a ORy y=f(x) S: Bounded by two curves y=f(x) and g(x) and between two vertical line y=a and y=b. S S = {( x, y ) | a ≤ x ≤ b, g ( x) ≤ y ≤ f ( x)} y=g(x) bo a b x A = ∫ [ f ( x ) − g ( x )]dx a
5. 5. Type 2 region : S: Lies between two curves x=f(y) and g(y) and between two line x=c and y=d. y y=d d S = {( x, y ) | c ≤ y ≤ d , g ( y ) ≤ x ≤ f ( y )} d x=g(y) x=f(y) A = ∫ [ f ( y ) − g ( y )]dy c S c y=c x o
6. 6. Type 3 region : modulus functions or functionswhen f>g in one part of interval and g>f in the otherpart To find the area between the curves y=f(x) and y=g(x) where f(x)>g(x) for some values of x but g(x)>f(x) for other values of x, then we split the given region into several regions S1, S2, …with areas A1, A2, ….,
7. 7. Type 3 region : modulus functions or functionswhen f>g in one part of interval and g>f in the otherpart To find the area between the curves y=f(x) and y=g(x) where f(x)>g(x) for some values of x but g(x)>f(x) for other values of x, then we split the given region into several regions S1, S2, …with areas A1, A2, …., s2 s3 π /4 π /2 s6 s4 s5
8. 8. TYPES OF QUESTIONSTYPE 1 AREA BOUNDED BETWEEN A CURVE AND A LINEEXAMPLE: Find the area bounded by the parabola x²=4y and the linex=4y-2 def int
9. 9. EXAMPLE 2 Find the area of the region {(x,y): x²≤y ≤x} def int
10. 10. TYPE 2 AREA BOUNDED BETWEEN TWO CURVES(a) Between 2 parabolasEXAMPLE: Find the area bounded by the parabola y²=4ax and x²=4ay,a>0 def int
11. 11. (b) Between a parabola and a circleEXAMPLE 1 : Find the area of the region {(x,y) : y² ≤ 4x, 4x²+4y² ≤ 9}
12. 12. EXAMPLE 2 : Find the area of the region {(x,y) : y² ≥ 4x, 4x²+4y² ≤ 9} def int
13. 13. (c) Between 2 circlesEXAMPLE: Find the area bounded by the circles x²+y²=1 and (x−1)²+y²=1
14. 14. (c) Between linesEXAMPLE: Using integration, find the area of triangle whosevertices are A(2,5) B(4,7) C(6,2)
15. 15. def int
16. 16. 5π / 4 3π / 2 π/ 2π /4