Evaluating an Integral With Geometry Formulas

  1. 1. The problem statement, all variables and given/known data

    I need to evaluate the definite integral of (x+2(1-x^2)^(1/2))dx from x=0 to x=1 using geometry formulas.

    2. Relevant equations

    None known.

    3. The attempt at a solution

    I'm actually trying to help one of my friends in AP Calculus with this problem. I know how to solve this with trigonometric substitutions, but they have not learned how to do these yet in their class. How do you use geometry formulas to solve something like this?
     
  2. jcsd
  3. [tex]\int (x+2\sqrt{1-x^2})dx[/tex]

    Correct? Geometry formulas? Haven't heard of that.
     
    Last edited: Jan 22, 2008
  4. Yeah, I need to evaluate that integral from x=0 to x=1.
     
  5. HallsofIvy

    HallsofIvy 40,297
    Staff Emeritus
    Science Advisor

    What, you've never heard of the integral being interpreted as the area under a curve?
    [tex]\int_0^1 x+ 2\sqrt{1- x^2} dx= \int_0^1 x dx+ 2\int_0^1 \sqrt{1- x^2}dx[/tex]

    The line y= x, along with y= 0 and x= 1 forms a triangle with base= 1 and height= 1. What is the area of that triangle?

    [itex]y= \sqrt{1- x^2}[/itex] is the upper half of [itex]x^2+ y^2= 1[/itex], a circle with radius 1. Multiplying by 2 just makes it the area of the entire circle. What is the area of that circle?

    This integral is the sum of the area of a triangle and the area of a circle.
     
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