Solve ∫ (from 0 to 2) of int_0^Lvdvdu

$$\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { L } v d v d u$$

Evaluate

$$L^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{0}^{L}v\mathrm{d}v\mathrm{d}u$$

Find the integral of \int _{0}^{L}v\mathrm{d}v using the table of common integrals rule \int a\mathrm{d}u=au.

$$\int _{0}^{L}v\mathrm{d}vu$$

Simplify.

$$\frac{L^{2}u}{2}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{2}L^{2}\times 2-\frac{1}{2}L^{2}\times 0$$

Simplify.

$$L^{2}$$

Differentiate w.r.t. L

$$2L$$