Solution: According to the meaning of the question, it can be concluded that the area of closed figure can be expressed by definite integral.

That is, the area =∫(lna, lnb)xdy,

And y=lnx, then x = e y.

Therefore, ∫ (lna, lnb) xdy = ∫ (lna, lnb) e ydy.

=e^y(lna,lnb)=e^lnb-e^lna=b-a。

That is, the area is b-a.

Extended data:

1, the properties of definite integral

If f(x) is the original function of F(x), then F(x)=∫f(x)dx. Then ∫(a, b)f(x)dx=F(b)-F(a)

When (1)a=b, then ∫(a, a)f(x)dx=F(a)-F(a)=0.

(2) when a ≠ b, then ∫ (a, b) f (x) dx =-∫ (b, a) f (x) dx = f (b)-f (a).

(3) ∫ (a, a) k * f (x) dx = k * ∫ (a, b) f (x) dx = k * (f (b)-f (a)), (where k is a non-zero constant)

2. The application of definite integral

(1) Solve the problem of finding the area of curved edges.

(2) Find the distance of linear motion with variable speed.

The distance S traveled by an object with variable speed and linear motion is equal to the definite integral of its velocity function v=v(t) (v(t)≥0) in the time interval [a, b].

(3) changing force to do work.

Under the action of variable force F=F(x), the work done by an object on the displacement interval [a, b] is equal to the definite integral of F=F(x) on [a, b].

3. Indefinite integral formula

∫ 1/(x^2)dx=- 1/x+C、∫adx=ax+C、∫ 1/xdx=ln|x|+C、∫e^xdx=e^x+C

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