Question

True or False. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{a}^{b} \sin x d x=\int_{a}^{b+2 \pi} \sin x d x $$

Short answer

True

Step-by-step solution

Recall the properties of the sine function

The sine function is a periodic function with a period of 2π. This means within every period, the amount of 'positive' area above the x-axis is the same as the amount of 'negative' area below the x-axis. Hence, if one takes an integral or the area under the curve of a sine function of one full period, it sums up to 0.

Calculate the left-hand side of the equation

The left-hand side of the equation is the integral of the sine function from 'a' to 'b'. There is no information provided about the relationship between 'a' and 'b', meaning they can have any real values. So, the integral value here can be anything and not necessarily zero.

Calculate the right-hand side of the equation

The right-hand side of the equation is the integral of the sine function from 'a' to 'b + 2π'. Since integral over each full period of the sin function (2π in width) sums up to zero, the integral value here would be the same as the integral from 'a' to 'b'.

Compare both sides

Step 2 and 3 show that the left-hand side and right-hand side of the equation have the same integral value. Hence, this confirms that the statement \(\int_{a}^{b} \sin x d x=\int_{a}^{b+2 \pi} \sin x d x\) is indeed true.