Solve ∫ (from 0 to 2) of x^2d0.2=

$$\int_{ 0 }^{ 2 } { x }^{ 2 } d 0.2 =$$

Evaluate

$$\frac{8d_{0}}{15}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{x^{2}d_{0}}{5}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\frac{d_{0}}{5}\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$\frac{d_{0}}{5}\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{d_{0}x^{3}}{15}$$

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.

$$\frac{1}{15}d_{0}\times 2^{3}-\frac{1}{15}d_{0}\times 0^{3}$$

Simplify.

$$\frac{8d_{0}}{15}$$

Differentiate w.r.t. d_0

$$\frac{8}{15} = 0.5333333333333333$$