Solve ∫ (from 0 to frac { pi) of {3}}6sinθdθ

$$\int _ { 0 } ^ { \frac { \pi } { 3 } } 6 \sin \theta d \theta$$

Evaluate

$$3$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 6\sin(\theta )\mathrm{d}\theta$$

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

$$6\int \sin(\theta )\mathrm{d}\theta$$

Use \int \sin(\theta )\mathrm{d}\theta =-\cos(\theta ) from the table of common integrals to obtain the result.

$$-6\cos(\theta )$$

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.

$$-6\cos(\frac{1}{3}\pi )+6\cos(0)$$

Simplify.

$$3$$