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# Calculus/Area by Integration

Question
Using Integrals, find the area of the region bounded by the tangent to the curve 4y=x^2 at the point (2,1) and the lines whose equations are x=2y and x=3y-3.

Three lines
Questioner:Akshar
Country:Gujarat, India
Category:Calculus
Private:No
Subject: Mathematics

Question:
Using Integrals, find the area of the region bounded by the tangent to the curve 4y=x^2 at the point (2,1) and the lines whose equations are x=2y and x=3y-3.
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Step 1:  Write  y = x^2/4;   y' = 2x/4 = x/2

At (2,1) m = dy/dx = 1,

Use the point-slope form of a line to show that the
equation of the tangent line is y = x - 1.

Step 2:  Draw the graphs of your three lines:

y = x - 1
y = x/2
y = (x+3)/3

(See attached)

Step 3: Solve three sets of simultaneous equations to find that the intersections are at  x = 2, x = 3, x= 6.

Step 4A: Integrate the difference between the red and green lines from x = 3 to x = 6.  <<< corrected

Step 4B: Integrate the difference between the red and black lines from x = 2 to x = 3.  <<< corrected

Calculus

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