You are here:

Calculus/Area by Integration

Advertisement


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.

Answer
Three lines
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.
.....................................

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

Step 5: Add.

Calculus

All Answers


Answers by Expert:


Ask Experts

Volunteer


Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

©2016 About.com. All rights reserved.