Easy Problem: Find Area of y = (x^5 + 4x^4)^(1/2) [-4,0]

[Sombra]

Easy problem... but urgent!

y = (x^5 + 4x^4)^(1/2)

They want me to find the area of this loop, and the boundaries are -4 and 0, but when I integrate it and plug in the -4, I get 0, which is clearly not the case. If you can help, I would love it.

 

[Tide]

I think you left out the second limit on the integration!

 

[WaR]

Can you post your integrating steps?
That integration is not equal to $$0$$
But
$$\int_{-4}^{0} \sqrt{x^{5}+4x^{4}} dx = \frac{2048}{105}$$

 

[Sombra]

I did:

Integration of (x^5 +4x^4)^(1/2) is 2/3 (x^5 +4x^4)^(3/2)

So 2/3(0 + 0)^(3/2) = 0

0 - [2/3 (-1024 + 1024)^(3/2) = 0

It doesn't work.

 

[Tide]

No, you integrated incorrectly!

 

[Sombra]

ok, the answer given in the back of the book is 4096/105. I know that even if that answer is not correct, it is more correct than mine because the area of that loop in the graph is clearly not 0.

 

[Sombra]

ok, I think the book is wrong because the calculator said that it was 19.50476233. I still don't know how this works though because... I still get 0.

 

[Pyrrhus]

Little hint

$$\int_{-4}^{0} \sqrt{x^{5}+4x^{4}} dx$$

$$\int_{-4}^{0} x^{2} \sqrt{x+4} dx$$

 

[Sombra]

that's how the original problem was set up. I distributed, thinking it would be easier. In any case, I get 0. Thanks

 

[Pyrrhus]

Post the original problem as the book states it.

 

[Sombra]

Sketch the graph y^2 = x^4(x+4) and find the area enclosed by the loop.

 

[Tide]

That is distinctly different from the problem you first posed - but your integration is still incorrect! :-)

 

[Sombra]

the boundaries are still -4 and 0, and my answer still comes out to be 0. What is correct? I guess I am not understanding. Please help!

 

[Tide]

$$\int x^2 \sqrt {x+4} dx = \frac {2 (x+4)^{3/2} (15 x^2 -48 x + 128)}{105}$$

 

[Sombra]

ok so 2/3(x+4)^3/2, I got that part, but how did you get the other part. I am not familiar with integrating something undistributed unless it follows the f(x) * f'(x) rule. Sorry for all the questions. Thank you so much!

 

[futb0l]

Use cyclovenom's hint

 

[Sombra]

Now I am totally lost... I know I have to integrate only the top half of this loop and multiply it by 2 to find the area or it will equal 0. I just do not know how to set up the problem to make it only the top half of the loop.

 

[Tide]

You can integrate by parts. You may get a slightly different form than I did because I did some simplification.