Evaluating an Integral With Geometry Formulas


by Frillth
Tags: evaluating, formulas, geometry, integral
Frillth
Frillth is offline
#1
Jan22-08, 11:46 PM
P: 80
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?
Phys.Org News Partner Science news on Phys.org
NASA's space station Robonaut finally getting legs
Free the seed: OSSI nurtures growing plants without patent barriers
Going nuts? Turkey looks to pistachios to heat new eco-city
rocomath
rocomath is offline
#2
Jan22-08, 11:48 PM
rocomath's Avatar
P: 1,757
[tex]\int (x+2\sqrt{1-x^2})dx[/tex]

Correct? Geometry formulas? Haven't heard of that.
Frillth
Frillth is offline
#3
Jan22-08, 11:51 PM
P: 80
Yeah, I need to evaluate that integral from x=0 to x=1.

HallsofIvy
HallsofIvy is offline
#4
Jan23-08, 06:33 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,882

Evaluating an Integral With Geometry Formulas


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.


Register to reply

Related Discussions
Help Evaluating an Integral Calculus & Beyond Homework 2
evaluating an integral Calculus & Beyond Homework 1
Evaluating an integral Calculus & Beyond Homework 5
Evaluating a cot integral Precalculus Mathematics Homework 3
Evaluating Integral Calculus & Beyond Homework 4