Finding area of a loop and under a curve

Short but useless answer -- yes, you can do this in Excel itself. The bad news is that Excel does not have a built in integration algorithm, so you'll either need to build your own algorithm or find something someone else has programmed.

If you choose to search for a third party program/add-in/algorithm, I expect you will find those designed with actual functions in mind (finding the area under polynomials for example), so you will probably need to adapt them to your tabulated data (or use some curve fitting techniques to get an equation that approximates the shape of each curve). Ultimately, this may not really be any easier than the way you are currently doing it using origin.

If you decide to build your own algorithm, here's how I would expect to proceed in a broad overview:

0) Review your calculus -- in particular the idea of Riemann Sums: https://en.wikipedia.org/wiki/Riemann_sum
https://calculushowto.com/problem-solving/riemann-sums/
1) Identify the point that separates the upper curve from the lower curve.
2) Using your chosen Riemann Sum, compute the areas under the upper and lower curves
3) The area between the curves is simply the difference of the two areas.

How would you like to proceed? What help can we offer in implementing your desired approach?

Left hand sum, right hand sum, or trapezoid (average of right hand and left hand sum)? The way I usually implement a Riemann sum in a spreadsheet like this is to compute x2-x1, then multiply by my desired estimate of the value of f. If I want a left hand sum, then it is y1*(x2-x1). If I want a right hand sum, it is y2*(x2-x1). If I want the trapezoidal sum, then it is AVERAGE(y2,y1)*(x2-x1). If I want something else, then I insert the appropriate value for y. Enter that into a column, copy the formula to 1 row before the bottom of the column. Then add a SUM() function somewhere convenient that sums that column.