Evaluate Integral \int_{-1}^1 (x^5 + 3) \sqrt{1 - x^2} dx

[JG89]

Homework Statement

Evaluate $$\int_{-1}^1 (x^5 + 3) \sqrt{1 - x^2} dx$$ without doing any computations.

Homework Equations

The Attempt at a Solution

$$\int_{-1}^1 (x^5 + 3) \sqrt{1 - x^2}dx = \int_{-1}^1 x^5 \sqrt{1-x^2} dx + \int_{-1}^1 3 \sqrt{1 - x^2}$$.

$$x^5 \sqrt{1 - x^2}$$ is an odd function and $$3 \sqrt{1 - x^2}$$ is an even function. So obviously $$\int_{-1}^1 x^5 \sqrt{1-x^2} dx = 0$$ and $$\int_{-1}^1 3 \sqrt{1-x^2} dx = 3 \int_{-1}^1 \sqrt{1-x^2} dx = 6 \int_0^1 \sqrt{1 - x^2} dx$$. And so:

$$\int_{-1}^1 (x^5 + 3) \sqrt{1 - x^2} dx = 6 \int_0^1 \sqrt{1 - x^2} dx$$.

If I am to do this without any computations, then this is the closest answer I could give (that I could think of). Is this answer acceptable?

 

[Dick]

I'm going to guess that they want you to handle the sqrt(1-x^2) by thinking about the geometry of the graph, and using a geometry formula.

 

[JG89]

Okay, I see what to do now. But wouldn't that still be a computation? Or do you think they mean don't do any contributions regarding the given integral?

 

[zcd]

$$\sqrt{1-x^{2}}$$ is the equation for a semicircle of radius 1 (should be self explanatory from here)

 

[Office_Shredder]

I think they mean don't use substitution or integration by parts to actually evaluate the integral. Think about what integration means, and what zcd said, and the answer pops out at you

 

[JG89]

Got'cha. Thanks guys

 

[Brisingr24]

is there a way u can include all the integral and square root symbols in here without using the ALT. (ALT way does not work with my qosmio.)??

 

[Mark44]

is there a way u can include all the integral and square root symbols in here without using the ALT. (ALT way does not work with my qosmio.)??

No one is using ALT characters, if that's what you mean. They are using LaTeX. Most likely your browser doesn't support it.

 

[Brisingr24]

im using the latest function of firefox and what is LaTex...??

 

[Mark44]

LaTeX is a scripting language that is used to text formatting, primarily for math and other scientific typesetting. With it, you can format integrals, limits, fractions, arrays, and many other things so that they appear as they do in textbooks.

I don't use Firefox, but it might be that there is a setting so that script isn't executed in Web pages. If so, that would probably prevent Firefox from rendering LaTeX script.