Question

In Exercises \(1-10\), evaluate the integral. $$ \int_{0}^{\cos y} y d x $$

Short answer

The evaluated integral is \( y\cos y \).

Step-by-step solution

Identify the function and limits

Here, the function to integrate is 'y' with respect to 'x'. The limits of the integral are from 0 to \( \cos y \).

Evaluate the integral

Using the definition of definite integrals, we want to find the antiderivative of the function and then evaluate it at the limits. The antiderivative of 'y' with respect to 'x' is \( yx \). The limits of the integral are specific values of 'x', therefore 'y' can be treated as a constant during integration. Then, using the limits to evaluate the integral - at \( \cos y \) and at 0 - we get \( y\cos y \) and 0, respectively.

Subtract the values

Subtracting these two values (as per the fundamental theorem of calculus), we get \( y\cos y - 0 = y\cos y \).