# Evaluating an Integral With Geometry Formulas

by Frillth
Tags: evaluating, formulas, geometry, integral
 P: 1,756 $$\int (x+2\sqrt{1-x^2})dx$$ Correct? Geometry formulas? Haven't heard of that.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,310 Evaluating an Integral With Geometry Formulas What, you've never heard of the integral being interpreted as the area under a curve? $$\int_0^1 x+ 2\sqrt{1- x^2} dx= \int_0^1 x dx+ 2\int_0^1 \sqrt{1- x^2}dx$$ 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? $y= \sqrt{1- x^2}$ is the upper half of $x^2+ y^2= 1$, 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.