MHB Who can give me a hint of how to calculate this integral,Thanks

[zhaojx84]

View attachment 7756
$$\int_{0}^{\frac{-a}{2}+\frac{\sqrt{392-{a}^{2}}}{2}} {y}^{2}\arcsin\left({\frac{a+y}{\sqrt{196-{y}^{2}}}}\right)\,dy$$

 

[tkhunny]

I'm not seeing an explicit evaluation, but I found the numerical solutions fascinating:View attachment 7762

 

[zhaojx84]

I'm not seeing an explicit evaluation, but I found the numerical solutions fascinating:

Thanks very much. Could you tell me which software did you use to express this curve? Is there a formula for this curve. Thanks again for your help.

 

[I like Serena]

Thanks very much. Could you tell me which software did you use to express this curve? Is there a formula for this curve. Thanks again for your help.

Hi zhaojx84, welcome to MHB! (Wave)

In GNU Octave, the free version of MatLab, we can do:

f = @(a, y) y^2 * asin((a + y) / (sqrt(196 - y^2)));
g = @(a) quad(@(y) f(a, y), 0, (-a / 2 + (sqrt(392 - a^2)) / 2));
x = -11:0.2:14;
y = arrayfun(g, x);
plot(x, y);

In Octave Online, we can quickly see what it does.

View attachment 7768

 

[tkhunny]

Thanks very much. Could you tell me which software did you use to express this curve? Is there a formula for this curve. Thanks again for your help.

I used PTC Mathcad Prime 3.0

It required pretty much what you see. Just type in what you want. The algebraic interpretation is provided.