You have seen how symmetry is used to simplify the integration of even and odd functions in Section 5.4. In this lab we extend these ideas and explore how more general symmetries are used to evaluate integrals. Specically, we focus on symmetry about a point and show how it can be applied to genuinely difcult integrals. In 1. Let's begin with a specic example. Consider the integral I sin2 xdx. Graph the integrand x) = sinzl 0 on the interval [0, ir/Z] and identify the region whose area is given by the integral. Draw the rectangle with opposite vertices at (0, 0) and (71/2, 1) and compute its area. Argue that the graph of x) = sinzx divides this rectangle in half and that the value of the integral is 7r/4. 2. The result of Step 1 depends on the assumption that the graph of y = f (x) divides the rectangle in question into two regions of equal area. The goal is to understand when this assumption can be made. Here is the key idea. A function f is said to symmetric about a point (p, q) if whenever the point (p x, q y) is on the graph of f then the point (p + x, q + y) is also on the graph. Said differently, f is symmetric about a point (p, q) if the line through the points (p, q) and (p + x, q + y) on the graph of y =f(x) intersects the graph at the point (p - x, q y). Show that a function symmetric about the point (p, q) satisfiesp x) +j(p + x) = Zp) for all x in the interval of interest