Solve ∫ (from x = 1 to 10) of 1/2x^2

$$\int _ { x = 1 } ^ { 10 } \frac { 1 } { 2 } x ^ { 2 }$$

Evaluate

$$\frac{333}{2}=166.5$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{x^{2}}{2}\mathrm{d}x$$

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

$$\frac{\int x^{2}\mathrm{d}x}{2}$$

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}.

$$\frac{x^{3}}{6}$$

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.

$$\frac{10^{3}}{6}-\frac{1^{3}}{6}$$

Simplify.

$$\frac{333}{2}$$