Question

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0}^{\pi / 2} \cos \left(\frac{2 x}{3}\right) d x $$

Short answer

The value of the definite integral is \( \frac{3}{2} \)

Step-by-step solution

Integrate the Function

The antiderivative of \( \cos(\frac{2x}{3}) \) is \( \frac{3\sin(\frac{2x}{3})}{2} \). Applying the Fundamental Theorem of calculus, the definite integral is computed by finding the antiderivative and evaluating it at the endpoints of the interval [0, \( \frac{\pi}{2} \)].

Evaluate the Antiderivative at the Endpoints

To evaluate the definite integral, substitute the endpoints into the antiderivative function: \( [\frac{3\sin(\frac{2}{3}\pi)}{2} - \frac{3\sin(\frac{2}{3}\cdot0)}{2}] \). This simplifies to \( \frac{3}{2} \).

Verify the Result with a Graphing Utility

To verify the result with a graphing utility, plot the integral of \( \cos(\frac{2x}{3}) \) from 0 to \( \frac{\pi}{2} \). The area under the curve should equal the value found in Step 2, thus confirming the result.