How to find area between curve and x-axis using trapz function?

x=0.6:0.01:1.8

y=sqrt((x.^2)-(sin(x.^4)))

plot(x,y)

How do I use the trapz function to find the area between y(x) and x-axis BETWEEN THE 2 LOCAL MINIMA of the curve?

Answer 1

[Star Strider]

I had some fun with this:

x=0.6:0.01:1.8;
y= @(x) sqrt((x.^2)-(sin(x.^4)));       % Anonymous Function
yx = y(x);                              % y(x)
dy = gradient(yx,x);                    % dy/dx
d2y = gradient(dy,x);                   % d^2y/dx^2
cs = dy.*circshift(dy,[0 -1]);          % Create Zero Crossings
xtr = find((cs <= 0) & (d2y > 0));      % Detect Curve Minima
ixrng = (xtr(1):xtr(2));                % X-Index Range Between Curve Minima
inty = trapz(yx(ixrng), x(ixrng));      % Integral

figure(1)
plot(x,yx)
hold on
patch([x(ixrng) x(flip(ixrng))], [yx(ixrng) zeros(size(ixrng))], [0.1 0.5 0.9])
hold off
grid
text(x(xtr(1))+0.25, yx(xtr(2))-0.5, sprintf('Integral = %8.5f',inty))

Plotting:

The calculus is straightforward, using the first derivative to find the extrema and the second derivative to determine minima. The ‘cs’ assignment creates zero-crossings, the ‘xtr’ assignment uses the first and second derivatives to detect the indices of the minima, and the rest is self-explanatory. The plot is just to demonstrate that the code does what it’s supposed to.

[instantglucose]

new to matlab and your code is clear and concise. thank you so much!

[instantglucose]

sorry but why is it trapz(y,x) and not trapz(x,y) ?

[Star Strider]

My pleasure!

The argument reversal in the trapz call is an oversight error on my part. I started out by doing numerical integration using the integral function (that has the function first and the limits second), then re-read your post and realised you wanted to use trapz.

It should be:

inty = trapz(x(ixrng), yx(ixrng));      % Integral

and the value of the integral is about 0.76. Nothing else in my code needs to change.

[instantglucose]

ok, i got it! thank you! :)

[Star Strider]

My pleasure!