Evaluating an Integral With Geometry Formulas

In summary, the person needs help with evaluating a definite integral using geometry formulas, specifically for the problem (x+2(1-x^2)^(1/2))dx from x=0 to x=1. They are not familiar with trigonometric substitutions and are wondering how to use geometry formulas to solve the problem. The integral can be interpreted as finding the area under a curve, and can be solved by finding the areas of a triangle and a circle.
  • #1
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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  • #2
[tex]\int (x+2\sqrt{1-x^2})dx[/tex]

Correct? Geometry formulas? Haven't heard of that.
 
Last edited:
  • #3
Yeah, I need to evaluate that integral from x=0 to x=1.
 
  • #4
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.
 

1. What is the purpose of using geometry formulas to evaluate an integral?

The purpose of using geometry formulas to evaluate an integral is to find the area under a curve on a graph. This can be done by using geometric shapes such as rectangles or triangles to approximate the area and then taking the limit as the number of shapes approaches infinity to get an exact value.

2. Can any integral be evaluated using geometry formulas?

No, not all integrals can be evaluated using geometry formulas. Only integrals that represent the area under a curve on a graph can be evaluated using geometry formulas. Other types of integrals, such as those involving trigonometric or exponential functions, require different methods to evaluate.

3. How do you use geometry formulas to evaluate an integral?

To use geometry formulas to evaluate an integral, you must first identify the shape that best approximates the area under the curve. Then, you divide the area into smaller, known shapes (such as rectangles or triangles) and use their corresponding formulas to calculate their individual areas. Finally, you add up all the areas of the smaller shapes to get an approximation of the area under the curve. Taking the limit as the number of shapes approaches infinity will give an exact value.

4. Are there any limitations to using geometry formulas to evaluate integrals?

Yes, there are limitations to using geometry formulas to evaluate integrals. One limitation is that it can only be used for certain types of integrals, as mentioned in the answer to question 2. Additionally, it can be time-consuming and may not always give an exact solution, as it relies on approximations.

5. Can geometry formulas be used to evaluate multidimensional integrals?

No, geometry formulas are not suitable for evaluating multidimensional integrals. The use of multiple dimensions makes it difficult to approximate the area using simple geometric shapes, and other methods such as the change of variables or numerical integration are required to evaluate these types of integrals.

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