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Calculus/Area between curves

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Expert: - 2/2/2010


Question
There is a line through the origin that divides the region bounded by the parabola y = 7 x - 8 x^2 and the x-axis into two regions with equal area. What is the slope of that line?

can you run me through the steps, I'm not sure what to do.

Answer

Half area.
Questioner: mike
Country: United States
Category: Calculus
Private: No
Subject: area
Question: There is a line through the origin that divides the region bounded by the parabola y = 7 x - 8 x^2 and the x-axis into two regions with equal area. What is the slope of that line?

can you run me through the steps, I'm not sure what to do.
..............................................

1. The parabola will intersect the x-axis at  x = 0 and  x = 7/8.

2. Integrate   7x - 8x^2  from 0 to 7/8.   That's your "whole area".  0.893

3. Your line has equation  y = mx.  [you want m].

--- Solve  mx = 7x - x^2 for x.  Your two solutions are  x = 0 and  x = (7-m)/8

4. Integrate   (7x - 8x^2) - (mx)  from  0 to (7-m)/8.

--- You get an answer that contains m.

5. Set that answer equal to "whole area" / 2 and solve for m.

--- I get something like   m = 1.444

--- Your intersection is  x = (7 - m)/8 = 0.6945

That should do it.

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.)

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