Problem 3. Consider the integral Using the area interpretation of the definite integral, convince yourself that the expression represents the area of a circle of radius R. (We have seen very similar examples in class and in Homework 1 and your first TUC contained such an integral.) We want to study the particular case when R = 3, i.e., x² dx. (a) (b) (1) to illustrate a special kind of substitution. Let x = 3 sin 0 and assume is between -7 π/2 and π/2. Find dx. Use substitution and determine new limits of integration to show 4 5²³ √9-x² dx : (c) R 4 S √R² - x² dx. 0 integral as (d) 3 4³. 36 √9 Hint. It may be helpful to remember the Pythagorean identity sin² (0) + cos² (0) = 1. Using the trigonometric identity cos² 0 = (1 + cos(20)), we can rewrite the new 3 [*1² Evaluate this definite integral. cos²0 de Cπ/2 5 cos² (0) de. JO = 36 = 18 π/2 S (1 + cos(20)) de. Did you find that the definite integral is 3² = 97, as expected?