Directions: Write up the following:

Consider the area under one arch of the curve

$y(t)=$asin$(\frac{1}{3}t)$

for $t0$ where $a$ is a positive constant.

A$)$ Sketch a graph of one arch of the curve $y(t)=$asin$(\frac{1}{3}t).$ Be sure to label the axis, the curve, and the end points of the arch. A brief explanation of how you determined the endpoints $($especially the larger one$)$ is required.

B$)$ Set up the definite integral needed to find the area.

C$)$ Make an appropriate substitution. Briefly describe why you chose to make this substitution. In addition, be sure to define the new variable, calculate the new differential, and demonstrate how the integrals value is the same when we change to the new variable.

D$)$ Evaluate the integral after the substitution. Be sure to show all important steps in using the Fundamental Theorem of Calculus, and to simplify your answer.