Evaluating an Integral With Geometry Formulas

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Homework Help Overview

The discussion revolves around evaluating the definite integral of the function (x + 2(1 - x^2)^(1/2)) from x=0 to x=1, with a focus on using geometric interpretations rather than traditional calculus methods.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to assist a friend by exploring how to evaluate the integral using geometry formulas, questioning how this can be achieved without trigonometric substitutions. Some participants express confusion about the concept of using geometry formulas for this integral.

Discussion Status

Participants are actively discussing the interpretation of the integral as an area under a curve. One participant outlines the geometric shapes involved, such as a triangle and a semicircle, suggesting a potential direction for evaluating the integral through area calculations. However, there is no explicit consensus on the approach yet.

Contextual Notes

There is mention of the original poster's friend being in an AP Calculus class, which may imply certain constraints on the methods they are familiar with, particularly the avoidance of trigonometric substitutions.

Frillth
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Homework Statement



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.

Homework Equations



None known.

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?
 
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\int (x+2\sqrt{1-x^2})dx

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

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