How do you find the area enclosed by the curves y equals e to the x and y equals x squared the y-axis and the line x equals 2?

Answer 1

Integrating y = f(x) between two limits gives you the area under that curve between the two limits and the x-axis:

area = ∫y dx

In this case, you have two limits of the y-axis which is the line x = 0, and the line x = 2

To find the area between the two curves, subtract the area under one from the area under the other.

Need to find area = ∫ f(x) - g(x) dx between the limits

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area = ∫e^x - x² dx between x = 0 and 2

= [e^x - x³/3] between x = 0 and 2

= (e² - 2³/3) - (e⁰ - 0³/3)

= e² - 8/3 - 1

= e² - 5/3

≈ 3.72