For example, you have created a plotted curve as below screenshot shown. This method will split the area between the curve and x axis to multiple trapezoids, calculate the area of every trapezoid individually, and then sum up these areas.

1. The first trapezoid is between x=1 and x=2 under the curve as below screenshot shown. You can calculate its area easily with this formula:  =(C3+C4)/2*(B4-B3).

2. Then you can drag the AutoFill handle of the formula cell down to calculate areas of other trapezoids.
Note: The last trapezoid is between x=14 and x=15 under the curve. Therefore, drag the AutoFill handle to the second to last cell as below screenshot shown.   

3. Now the areas of all trapezoids are figured out. Select a blank cell, type the formula =SUM(D3:D16) to get the total area under the plotted area.

This method will use the chart trendline to get an equation for the plotted curve, and then calculate area under the plotted curve with the definite integral of the equation.

1. Select the plotted chart, and click Design (or Chart Design) > Add Chart Element > Trendline > More Trendline Options. See screenshot:

2. In the Format Trendline pane:
(1) In the Trendline Options section, choose one option which is most matched with your curve;
(2) Check the Display Equation on chart option.

3. Now the equation is added into the chart. Copy the equation into your worksheet, and then get the definite integral of the equation.

In my case, the equation general by trendline is y = 0.0219x^2 + 0.7604x + 5.1736, therefore its definite integral is F(x) = (0.0219/3)x^3 + (0.7604/2)x^2 + 5.1736x + c.

4. Now we plug in the x=1 and x=15 to the definite integral, and calculate the difference between both calculations results. The difference represents the area under the plotted curve.

Area = F(15)-F(1)
Area =(0.0219/3)*15^3+(0.7604/2)*15^2+5.1736*15-(0.0219/3)*1^3-(0.7604/2)*1^2-5.1736*1
Area = 182.225