Evaluate the following integral as a limit of sums: ∫ e^x dx, x ∈ [0, 2]

Question

Evaluate the following integral as a limit of sums:

\(\int\limits_{0}^{2} \)e^x dx

Answer 1

To find:  \(\int\limits_{0}^{2} \)e^x dx

Formula used:

where,

Here, f(x) = e^x and a = 0

Now, by putting x = 0 in f(x) we get,

f(0) = e⁰ = 1

f(h)

= (e)^h

= e^h

Similarly, f(2h)

= e^2h

This is G.P. (Geometric Progression) of n terms whose first term(a) is 1.

and common ratio(r) = \(\cfrac{e^h}1\) = e^h

Sum of n terms of a G.P. is given by,

⇒ I = e² - 1

Hence, the value of 

 \(\int\limits_{0}^{2} \)e^x dx = e² - 1